Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602022-11-17T18:59:28.764376ZUnknown authorWerkzeugPrefacehttps://zbmath.org/1496.000692022-11-17T18:59:28.764376ZFrom the text: The Tenth International Conference on Matrix-Analytic Methods in Stochastic Models (MAM10) was held at the University of Tasmania in Hobart from the 13th to the 15th of February 2019.
This issue of Stochastic Models, which is dedicated to Matrix-Analytic Methods, provides a sample of the research presented at MAM10.
The seven papers contained in this issue cover a broad range of research in the areas of quasi-birth-and-death processes, phase-type and matrix exponential distributions, continuous-state Markov processes, and applications.A lifetime of excursions through random walks and Lévy processeshttps://zbmath.org/1496.010092022-11-17T18:59:28.764376Z"Chaumont, Loïc"https://zbmath.org/authors/?q=ai:chaumont.loic"Kyprianou, Andreas E."https://zbmath.org/authors/?q=ai:kyprianou.andreas-eSummary: We recall the many highlights of Professor Ron Doney's career summarising his main contributions to the theory of random walks and Lévy processes.
For the entire collection see [Zbl 1478.60005].Relatively exchangeable structureshttps://zbmath.org/1496.031602022-11-17T18:59:28.764376Z"Crane, Harry"https://zbmath.org/authors/?q=ai:crane.harry"Towsner, Henry"https://zbmath.org/authors/?q=ai:towsner.henrySummary: We study random relational structures that are \textit{relatively exchangeable} -- that is, whose distributions are invariant under the automorphisms of a reference structure \(\mathfrak{M}\). When \(\mathfrak{M}\) is \textit{ultrahomogeneous} and has \textit{trivial definable closure}, all random structures relatively exchangeable with respect to \(\mathfrak{M}\) satisfy a general Aldous-Hoover-type representation. If \(\mathfrak{M}\) also satisfies the \textit{n-disjoint amalgamation property} (\(n\)-DAP) for all \(n \geq 1\), then relatively exchangeable structures have a more precise description whereby each component depends locally on \(\mathfrak{M}\).Relative exchangeability with equivalence relationshttps://zbmath.org/1496.031612022-11-17T18:59:28.764376Z"Crane, Harry"https://zbmath.org/authors/?q=ai:crane.harry"Towsner, Henry"https://zbmath.org/authors/?q=ai:towsner.henrySummary: We describe an Aldous-Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to \textit{N. Ackerman} [``Representations of \(\mathrm{Aut}(\mathcal{M})\)-invariant measures. I'', Preprint, \url{arXiv:1509.06170}] and \textit{H. Crane} and \textit{H. Towsner} [J. Symb. Log. 83, No. 2, 416--442 (2018; Zbl 1496.03160)] and hierarchical exchangeability results due to \textit{T. Austin} and \textit{D. Panchenko} [Probab. Theory Relat. Fields 159, No. 3--4, 809--823 (2014; Zbl 1306.60027)].Geometric random graphs on circleshttps://zbmath.org/1496.051152022-11-17T18:59:28.764376Z"Angel, Omer"https://zbmath.org/authors/?q=ai:angel.omer"Spinka, Yinon"https://zbmath.org/authors/?q=ai:spinka.yinonSummary: Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference \(L\), and find a surprising dependency of behaviour on the rationality of \(L\).
For the entire collection see [Zbl 1478.60006].Concentration inequalities from monotone couplings for graphs, walks, trees and branching processeshttps://zbmath.org/1496.051762022-11-17T18:59:28.764376Z"Johnson, Tobias"https://zbmath.org/authors/?q=ai:johnson.tobias"Peköz, Erol"https://zbmath.org/authors/?q=ai:pekoz.erol-aSummary: Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. \textit{E. A. Peköz} et al. [Ann. Probab. 44, No. 3, 1776--1816 (2016; Zbl 1367.60019)] exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to concentration inequalities for preferential attachment random graphs, branching processes, random walk local time statistics and the size of random subtrees of uniformly random binary rooted plane trees.Cutoff at the entropic time for random walks on covered expander graphshttps://zbmath.org/1496.051772022-11-17T18:59:28.764376Z"Bordenave, Charles"https://zbmath.org/authors/?q=ai:bordenave.charles"Lacoin, Hubert"https://zbmath.org/authors/?q=ai:lacoin.hubertSummary: It is a fact simple to establish that the mixing time of the simple random walk on a \(d\)-regular graph \(G_n\) with \(n\) vertices is asymptotically bounded from below by \(\frac{d}{d-2} \frac{\log n}{\log (d-1)}\). Such a bound is obtained by comparing the walk on \(G_n\) to the walk on \(d\)-regular tree \(\mathcal{T}_d\). If one can map another transitive graph \(\mathcal{G}\) onto \(G_n\), then we can improve the strategy by using a comparison with the random walk on \(\mathcal{G}\) (instead of that of \(\mathcal{T}_d\)), and we obtain a lower bound of the form \(\frac{1}{\mathfrak{h}}\log n\), where \(\mathfrak{h}\) is the entropy rate associated with \(\mathcal{G}\). We call this the entropic lower bound.
It was recently proved that in the case \(\mathcal{G} =\mathcal{T}_d\), this entropic lower bound (in that case \(\frac{d}{d-2} \frac{\log n}{\log (d-1)}\)) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on \(G_n\) under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random \(d\)-regular graphs and to random walks on random \(n\)-lifts of a base graph (including nonreversible walks).The local weak limit of \(k\)-dimensional hypertreeshttps://zbmath.org/1496.051942022-11-17T18:59:28.764376Z"Mészáros, András"https://zbmath.org/authors/?q=ai:meszaros.andrasSummary: Let \(\mathcal{C}(n,k)\) be the set of \(k\)-dimensional simplicial complexes \(C\) over a fixed set of \(n\) vertices such that:
\par i) \(C\) has a complete \(k-1\)-skeleton;
\par ii) \(C\) has precisely \(\binom{n-1}{k}\) \(k\)-faces;
\par iii) the homology group \(H_{k-1}(C)\) is finite.
Consider the probability measure on \(\mathcal{C}(n,k)\) where the probability of a simplicial complex \(C\) is proportional to \(|H_{k-1}(C)|^2\). For any fixed \(k\), we determine the local weak limit of these random simplicial complexes as \(n\) tends to infinity. This local weak limit turns out to be the same as the local weak limit of the 1-out \(k\)-complexes investigated by \textit{N. Linial} and \textit{Y. Peled} [Ann. Math. (2) 184, No. 3, 745--773 (2016; Zbl 1348.05193)].Extrema of Lüroth digits and a zeta function limit relationhttps://zbmath.org/1496.111042022-11-17T18:59:28.764376Z"Athreya, Jayadev S."https://zbmath.org/authors/?q=ai:athreya.jayadev-s"Athreya, Krishna B."https://zbmath.org/authors/?q=ai:athreya.krishna-balasundaramThe authors consider some problems of digit distribution for Lüroth expansions, i.e., the number theoretic expansions generated by the map \(L: (0,1] \rightarrow (0,1]\) given by \(L(x) = N(x)((N(x)+1)x-1)\) with \(N(x) = \lfloor 1/x \rfloor\). The Lüroth digits of \(x\) are the integers \(N(L^{k-1} (x))\), and if \(U\) is a uniformly distributed random variable on \((0,1]\), the random variables \(X_k = N(T^{k-1}(U))\) are independent, identically distributed (IID) with \(\mathbb{P}(X_1 = n) = \frac{1}{n(n+1)}\).
Letting \(\rho_k\) denote the probability that the value of \(M_k = \max\{X_1, \dots, X_k\}\) is attained exactly once among \(X_1, \dots, X_k\), the authors prove that \[ \rho_k = k \left(2^{k-1} + \sum_{j=2}^k T(k-1,k-j) \zeta(j) (-1)^{j+1}\right), \] where \(\zeta(j)\) is the Riemann \(\zeta\)-function evaluated at \(j\) and \(T(l,j) = \sum_{i=0}^j \binom{l}{i}\). It is known that in this particular setting, \(\lim_{k \rightarrow \infty} \rho_k = 1\), so the limiting value of the term on the right hand side must be equal to \(1\).
Subsequently, in the spirit of \textit{H. G. Diamond} and \textit{J. D. Vaaler} [Pac. J. Math. 122, 73--82 (1986; Zbl 0589.10056)], it is shown that with probability \(1\), \[ \lim_{k\rightarrow \infty} \frac{S_k - M_k}{k \log k} = 1, \] where \(S_k = \sum_{i=1}^k X_i\). The result of [loc. cit.] is the corresponding result for continued fractions, which is considerably more difficult to prove, as the digits in this case are not IID. The authors conjecture that for continued fractions, the probability of having a unique largest digit should still converge to \(1\).
Reviewer: Simon Kristensen (Aarhus)General tail bounds for random tensors summation: majorization approachhttps://zbmath.org/1496.150192022-11-17T18:59:28.764376Z"Chang, Shih Yu"https://zbmath.org/authors/?q=ai:chang.shih-yu"Wei, Yimin"https://zbmath.org/authors/?q=ai:wei.yimin.1|wei.yiminSummary: In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this research extends previous work by considering the tensors summation tail behavior of the top \(k\)-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in [\textit{S. Y. Chang}, ``Convenient tail bounds for sums of random tensors'', Preprint, \url{arXiv:2012.15428}]. Majorization and antisymmetric tensor product tools are main techniques utilized to establish inequalities for unitarily invariant norms of multivariate tensors. The Laplace transform method is integrated with these inequalities for unitarily invariant norms of multivariate tensors to give us tail bounds estimation for the Ky Fan \(k\)-norm for a function of the tensors summation. By restricting different random tensor conditions, we obtain generalized tensor Chernoff and Bernstein inequalities.On the condition number of the shifted real Ginibre ensemblehttps://zbmath.org/1496.150252022-11-17T18:59:28.764376Z"Cipolloni, Giorgio"https://zbmath.org/authors/?q=ai:cipolloni.giorgio"Erdös, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schröder, Dominik"https://zbmath.org/authors/?q=ai:schroder.dominikHypergeometry, integrability and Lie theory. Virtual conference, Lorentz Center, Leiden, the Netherlands, December 7--11, 2020https://zbmath.org/1496.170012022-11-17T18:59:28.764376ZPublisher's description: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7--11, 2020, which was dedicated to the 50th birthday of Jasper Stokman.
The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Etingof, Pavel; Kazhdan, David}, Characteristic functions of \(p\)-adic integral operators, 1-27 [Zbl 07602313]
\textit{Garbali, Alexandr; Zinn-Justin, Paul}, Shuffle algebras, lattice paths and the commuting scheme, 29-68 [Zbl 07602314]
\textit{Kolb, Stefan}, The bar involution for quantum symmetric pairs -- hidden in plain sight, 69-77 [Zbl 07602315]
\textit{Koornwinder, Tom H.}, Charting the \(q\)-Askey scheme, 79-94 [Zbl 07602316]
\textit{Rains, Eric M.}, Filtered deformations of elliptic algebras, 95-154 [Zbl 07602317]
\textit{Regelskis, Vidas; Vlaar, Bart}, Pseudo-symmetric pairs for Kac-Moody algebras, 155-203 [Zbl 07602318]
\textit{Reshetikhin, N.; Stokman, J. V.}, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, 205-241 [Zbl 07602319]
\textit{Rösler, Margit; Voit, Michael}, Elementary symmetric polynomials and martingales for Heckman-Opdam processes, 243-262 [Zbl 07602320]
\textit{Schomerus, Volker}, Conformal hypergeometry and integrability, 263-285 [Zbl 07602321]
\textit{Varchenko, Alexander}, Determinant of \(\mathbb{F}_p\)-hypergeometric solutions under ample reduction, 287-307 [Zbl 07602322]
\textit{Varchenko, Alexander}, Notes on solutions of KZ equations modulo \(p^s\) and \(p\)-adic limit \(s\to\infty\), 309-347 [Zbl 07602323]Martin boundary of killed random walks on isoradial graphshttps://zbmath.org/1496.310062022-11-17T18:59:28.764376Z"Boutillier, Cédric"https://zbmath.org/authors/?q=ai:boutillier.cedric"Raschel, Kilian"https://zbmath.org/authors/?q=ai:raschel.kilianThis article discusses killed planar random walks on isoradial graphs. Unlike the lattice case, isoradial graphs present some difficulties such as: not translation invariant, do not admit any group structure and are spatially non-homogeneous. In the current work, the authors compute the asymptotics of the Martin kernel, deduce the Martin boundary and show its minimality.
Reviewer: Marius Ghergu (Dublin)Limit theorems for Jacobi ensembles with large parametershttps://zbmath.org/1496.330102022-11-17T18:59:28.764376Z"Hermann, Kilian"https://zbmath.org/authors/?q=ai:hermann.kilian"Voit, Michael"https://zbmath.org/authors/?q=ai:voit.michaelSummary: Consider \(\beta\)-Jacobi ensembles on the alcoves
\[
A:=\{ x\in\mathbb{R}^N \mid -1\leq x_1\leq \cdots\leq x_N\leq 1\}
\]
with parameters \(k_1,k_2,k_3\geq 0\). In the freezing case \((k_1,k_2,k_3)=\kappa\cdot (a,b,1)\) with \(a,b>0\) fixed and \(\kappa\to\infty\), we derive a central limit theorem. The drift and covariance matrix of the limit are expressed via the zeros of classical Jacobi polynomials. We also determine the eigenvalues and eigenvectors of the covariance matrices. Our results are related to corresponding limits for \(\beta\)-Hermite and Laguerre ensembles for \(\beta\to\infty\).Impulsive conformable fractional stochastic differential equations with Poisson jumpshttps://zbmath.org/1496.340802022-11-17T18:59:28.764376Z"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-mSummary: In this article, periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed. By using stochastic analysis, fractional calculus, Doob's martingale inequality and Cauchy-Schwarz inequality, we show that the solution of the conformable fractional impulsive stochastic differential equations with Poisson jumps converges to the corresponding averaged conformable fractional stochastic differential equations with Poisson jumps and without impulses.Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delayshttps://zbmath.org/1496.341222022-11-17T18:59:28.764376Z"Jin, Shoubo"https://zbmath.org/authors/?q=ai:jin.shouboSummary: The Picard iteration method is used to study the existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays. Several sufficient conditions are specified to ensure that the equation has a unique solution. First, the stochastic Volterra-Levin equation is transformed into an integral equation. Then, to obtain the solution of the integral equation, the successive approximation sequences are constructed, and the existence and uniqueness of solutions for the stochastic Volterra-Levin equation are derived by the convergence of the sequences. Finally, two examples are given to demonstrate the validity of the theoretical results.Homogenization of stochastic conservation laws with multiplicative noisehttps://zbmath.org/1496.350422022-11-17T18:59:28.764376Z"Frid, Hermano"https://zbmath.org/authors/?q=ai:frid.hermano"Karlsen, Kenneth H."https://zbmath.org/authors/?q=ai:karlsen.kenneth-hvistendahl"Marroquin, Daniel"https://zbmath.org/authors/?q=ai:marroquin.daniel-rSummary: We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.Ill posedness for the full Euler system driven by multiplicative white noisehttps://zbmath.org/1496.352942022-11-17T18:59:28.764376Z"Chiodaroli, Elisabetta"https://zbmath.org/authors/?q=ai:chiodaroli.elisabetta"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Flandoli, Franco"https://zbmath.org/authors/?q=ai:flandoli.francoThe paper under review establishes the ill-posedness of weak solutions (more specifically, the existence of infinitely many global-in-time weak solutions) to the full Euler system describing the motion of a compressible fluid driven by a multiplicative white noise in dimension 2 or 3. In [\textit{D. Breit} et al., Anal. PDE 13, No. 2, 371--402 (2020; Zbl 1435.35289)], it was shown that the isentropic Euler system driven by a general additive/multiplicative white noise is ill-posed, but the infinitely many weak solutions constructed therein may be physically irrelevant, in the sense that they may experience an initial energy jump. In contrast, the infinitely many weak solutions shown to exist in this paper are physical admissible -- they conserve total energy and satisfy an entropy inequality.
Key ingredients of the proof include transforming the problem into PDE with random parameters, applying a result of \textit{C. De Lellis} and \textit{L. Székelyhidi jun.} [Arch. Ration. Mech. Anal. 195, No. 1, 225--260 (2010; Zbl 1192.35138)] for incompressible Euler with constant pressure obtained via convex integration, and employing a pasting argument for the particular solutions obtained as above. The strategy of the arguments follows essentially \textit{T. Luo} et al. [Adv. Math. 291, 542--583 (2016; Zbl 1337.35114)].
Reviewer: Siran Li (Shanghai)Solution properties of the incompressible Euler system with rough path advectionhttps://zbmath.org/1496.352952022-11-17T18:59:28.764376Z"Crisan, Dan"https://zbmath.org/authors/?q=ai:crisan.dan-o"Holm, Darryl D."https://zbmath.org/authors/?q=ai:holm.darryl-d"Leahy, James-Michael"https://zbmath.org/authors/?q=ai:leahy.james-michael"Nilssen, Torstein"https://zbmath.org/authors/?q=ai:nilssen.torstein-kSummary: The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work [J. Nonlinear Sci. 29, No. 3, 813--870 (2019; Zbl 1433.60051); Adv. Math. 404, Part A, Article ID 108409, 75 p. (2022; Zbl 1495.35145)], we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in \(L^2\)-Sobolev spaces \(H^m\) with integer regularity \(m \geq \lfloor d / 2 \rfloor + 2\) and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the \(L_t^1 L_x^\infty\)-norm of the vorticity. In dimension two, we show that the \(L^p\)-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.On nonlinear Schrödinger equations with random initial datahttps://zbmath.org/1496.353592022-11-17T18:59:28.764376Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitiaSummary: This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.Retraction note: ``Existence and uniqueness of solutions for the Schrödinger integrable boundary value problem''https://zbmath.org/1496.353702022-11-17T18:59:28.764376Z"Wang, Jianjie"https://zbmath.org/authors/?q=ai:wang.jianjie"Mai, Ali"https://zbmath.org/authors/?q=ai:mai.ali"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong|wang.hong.1|wang.hong.4|wang.hong.3|wang.hong.2|wang.hong.7|wang.hong.5From the text: The Editors-in-Chief have retracted the article [ibid. 2018, Paper No. 74, 13 p. (2018; Zbl 07509530)] by \textit{J. Wang} et al. because it shows evidence of peer review manipulation. In addition, it overlaps significantly with an article by \textit{J. Sun} et al. [J. Inequal. Appl. 2018, Paper No. 100, 15 p. (2018; Zbl 1496.60075)] that was simultaneously under consideration with another journal. The authors have not responded to any correspondence from the publisher regarding this retraction.Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspectivehttps://zbmath.org/1496.353892022-11-17T18:59:28.764376Z"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Wang, Ziheng"https://zbmath.org/authors/?q=ai:wang.ziheng"Zhang, Yuan"https://zbmath.org/authors/?q=ai:zhang.yuan"Zhou, Zhennan"https://zbmath.org/authors/?q=ai:zhou.zhennanCorrection to: ``Second order local minimal-time mean field games''https://zbmath.org/1496.353912022-11-17T18:59:28.764376Z"Ducasse, Romain"https://zbmath.org/authors/?q=ai:ducasse.romain"Mazanti, Guilherme"https://zbmath.org/authors/?q=ai:mazanti.guilherme"Santambrogio, Filippo"https://zbmath.org/authors/?q=ai:santambrogio.filippoCorrection to the authors' paper [ibid. 29, No. 4, Paper No. 37, 32 p. (2022; Zbl 1492.35352)].Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graphhttps://zbmath.org/1496.354052022-11-17T18:59:28.764376Z"Erbar, Matthias"https://zbmath.org/authors/?q=ai:erbar.matthias"Forkert, Dominik"https://zbmath.org/authors/?q=ai:forkert.dominik"Maas, Jan"https://zbmath.org/authors/?q=ai:maas.jan"Mugnolo, Delio"https://zbmath.org/authors/?q=ai:mugnolo.delioSummary: This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan-Kinderlehrer-Otto, we show that McKean-Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.Higher-order regularity of the free boundary in the inverse first-passage problemhttps://zbmath.org/1496.354592022-11-17T18:59:28.764376Z"Chen, Xinfu"https://zbmath.org/authors/?q=ai:chen.xinfu"Chadam, John"https://zbmath.org/authors/?q=ai:chadam.john-m"Saunders, David"https://zbmath.org/authors/?q=ai:saunders.david-claudeOn weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effectshttps://zbmath.org/1496.354662022-11-17T18:59:28.764376Z"Medjo, T. Tachim"https://zbmath.org/authors/?q=ai:tachim-medjo.theodoreSummary: We consider a stochastic Allen-Cahn-Navier-Stokes equations with inertial effects in a bounded domain \(D\subset\mathbb{R}^d\), \(d = 2, 3\), driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo-Galerkin approximation, compactness method and the Skorokhod representation theorem.Deep neural network surrogates for nonsmooth quantities of interest in shape uncertainty quantificationhttps://zbmath.org/1496.354672022-11-17T18:59:28.764376Z"Scarabosio, Laura"https://zbmath.org/authors/?q=ai:scarabosio.lauraPropagation of stochastic travelling waves of cooperative systems with noisehttps://zbmath.org/1496.354692022-11-17T18:59:28.764376Z"Wen, Hao"https://zbmath.org/authors/?q=ai:wen.hao"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Li, Yuhong"https://zbmath.org/authors/?q=ai:li.yuhongSummary: We consider the cooperative system driven by a multiplicative Itô type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.Optimal stochastic forcings for sensitivity analysis of linear dynamical systemshttps://zbmath.org/1496.370062022-11-17T18:59:28.764376Z"Nechepurenko, Yuri M."https://zbmath.org/authors/?q=ai:nechepurenko.yuri-m"Zasko, Grigory V."https://zbmath.org/authors/?q=ai:zasko.grigory-vSummary: The paper is devoted to the construction of optimal stochastic forcings for studying the sensitivity of linear dynamical systems to external perturbations. The optimal forcings are sought to maximize the Schatten norms of the response. As an example, we consider the problem of constructing the optimal stochastic forcing for the linear dynamical system arising from the analysis of large-scale structures in a stratified turbulent Couette flow.Random attractors for dissipative systems with rough noiseshttps://zbmath.org/1496.370512022-11-17T18:59:28.764376Z"Duc, Luu Hoang"https://zbmath.org/authors/?q=ai:duc.luu-hoangThe author studies the long-term behavior of the dynamics generated by rough differential equations with Hölder driving noises. The existence and upper semi-continuity of the global pullback attractor for certain dissipative systems with noises is proved. Further, it is shown that if one starts with a strictly disspative system, then the random attractor is a singleton if the perturbations possess sufficiently small noise intensity.
Reviewer: Marks Ruziboev (Wien)Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noisehttps://zbmath.org/1496.370552022-11-17T18:59:28.764376Z"Lv, Xiang"https://zbmath.org/authors/?q=ai:lv.xiangSummary: This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \(f\) is monotone (or anti-monotone) and the global Lipschitz constant of \(f\) is smaller than the positive real part of the principal eigenvalue of the competitive matrix \(A\), the random dynamical system (RDS) generated by SDEs has an unstable \(\mathscr{F}_+\)-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \(\mathscr{F}_+ = \sigma \{ \omega \mapsto W_t (\omega):t\geq 0\}\) is the future \(\sigma\)-algebra. In addition, we get that the \(\alpha\)-limit set of all pull-back trajectories starting at the initial value \(x(0) = x\in\mathbb{R}^n\) is a single point for all \(\omega\in\Omega\), i.e., the unstable \(\mathscr{F}_+\)-measurable random equilibrium. Applications to stochastic neural network models are given.Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equationshttps://zbmath.org/1496.370812022-11-17T18:59:28.764376Z"Kinra, K."https://zbmath.org/authors/?q=ai:kinra.kush"Mohan, M. T."https://zbmath.org/authors/?q=ai:mohan.manil-tExit versus escape for stochastic dynamical systems and application to the computation of the bursting time duration in neuronal networkshttps://zbmath.org/1496.370822022-11-17T18:59:28.764376Z"Zonca, Lou"https://zbmath.org/authors/?q=ai:zonca.lou"Holcman, David"https://zbmath.org/authors/?q=ai:holcman.davidSummary: We study the exit time of two-dimensional dynamical systems perturbed by a small noise that exhibits two peculiar behaviors: (1) The maximum of the probability density function of trajectories is not located at the point attractor. The distance between the maximum and the attractor increases with the noise amplitude \(\sigma\), as shown by using WKB approximation and numerical simulations. (2) For such systems, exiting from the basin of attraction is not sufficient to guarantee a full escape, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. We apply these results to study neuronal networks that can generate bursting events. To analyze interburst durations and their statistics, we study the phase space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics. We find that the interburst corresponds to an escape with multiple reentries inside the basin of attraction. To conclude, escaping far away from a basin of attraction is not equivalent to reaching the boundary, thus providing an explanation for non-Poissonian long interburst durations present in neuronal dynamics.Stationary distribution, extinction and probability density function of a stochastic vegetation-water model in arid ecosystemshttps://zbmath.org/1496.370972022-11-17T18:59:28.764376Z"Zhou, Baoquan"https://zbmath.org/authors/?q=ai:zhou.baoquan"Han, Bingtao"https://zbmath.org/authors/?q=ai:han.bingtao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqing"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawar"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmedSummary: In this paper, we study a three-dimensional stochastic vegetation-water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution \(\varpi (\cdot)\). In a biological interpretation, the existence of the distribution \(\varpi (\cdot)\) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium \(\overline{D}^{\ast}\) related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker-Planck equation, we obtain the approximate expression of the distribution \(\varpi (\cdot)\) around the equilibrium \(\overline{D}^{\ast}\). In addition, we obtain sufficient condition \(\mathscr{R}_0^E <1\) for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics.AP-frames and stationary random processeshttps://zbmath.org/1496.420072022-11-17T18:59:28.764376Z"Centeno, Hernán D."https://zbmath.org/authors/?q=ai:centeno.hernan-d"Medina, Juan M."https://zbmath.org/authors/?q=ai:medina.juan-miguelSummary: It is known that, in general, an AP-frame is an \(L^2 (\mathbb{R})\)-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system \(\{ g(t - k) e^{il(t-k)}, l \in \mathbb{L} = \omega_0 \mathbb{Z}, k \in \mathbb{K} = t_0 \mathbb{Z}\}\) to be an \(L^2 (\mathbb{R})\)-Frame in terms of Gaussian stationary random processes. In addition, if \(X = (X(t))_{t \in \mathbb{R}}\) is a wide sense stationary random process, we study density conditions for the associated stationary sequences \(\{\langle X, g_{k, l} \rangle, l \in \mathbb{L}, k \in \mathbb{K}\}\).Hardy-Littlewood maximal operator on variable Lebesgue spaces with respect to a probability measurehttps://zbmath.org/1496.420252022-11-17T18:59:28.764376Z"Moreno, Jorge"https://zbmath.org/authors/?q=ai:moreno.jorge"Pineda, Ebner"https://zbmath.org/authors/?q=ai:pineda.ebner"Rodriguez, Luz"https://zbmath.org/authors/?q=ai:rodriguez.luz"Urbina, Wilfredo O."https://zbmath.org/authors/?q=ai:urbina-romero.wilfredo-oIn this paper, the authors established the strong and weak boundedness of Hardy-Littlewood maximal operators on variable Lebesgue spaces \(L^{p(\cdot)}(\mu)\) with respect to a probability Borel measure \(\mu\) for two conditions of regularity on the exponent function \(p(\cdot)\).
To be more precise, let \(\mu\) be a Radon measure and \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), The Hardy-Littlewood non-centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu f(x)=\sup_{B\ni x}\fint_{B}|f(x)|\mu(dy),
\]
and the Hardy-Littlewood centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu^c f(x)=\sup_{r>0}\fint_{B(x,r)}|f(y)|\mu(dy),
\]
The authors first prove the boundedness with the condition \(\mathcal{P}^0_\mu(\mathbb{R}^d)\).
Theorem 1. Let \(p(\cdot)\in\mathcal{P}^0_\mu(\mathbb{R}^d)\) be continuous with \(p_- > 1\).
(i) There exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
(ii) If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Then, they gave the boundedness with the condition \(\mathcal{P}_\mu(\mathbb{R}^d)\).
Theorem 2. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) with \(p_- > 1\) be such that \(1/{p(\cdot)}\) is continuous. If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Theorem 3. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) be such that \(1/{p(\cdot)}\) is continuous, then there exists \(C>0\) depending on \(p\) such that
\[
\|t\chi_{\{x\in\mathbb{R}^d:M_\mu^c f(x)>t\}}\|_{p(\cdot),\mu}\leq C\|f\|_{p(\cdot),\mu}
\]
for all \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), \(t>0\).
They also extended some properties of this operator to a probability Borel measure \(\mu\), the key to extending these results is using the Besicovitch covering lemma instead of the Calderón Zygmund decomposition.
Reviewer: Qingying Xue (Beijing)Dimension independent atomic decomposition for dyadic martingale \(\mathbb{H}^1\)https://zbmath.org/1496.420262022-11-17T18:59:28.764376Z"Paluszynski, Maciej"https://zbmath.org/authors/?q=ai:paluszynski.maciej"Zienkiewicz, Jacek"https://zbmath.org/authors/?q=ai:zienkiewicz.jacekSummary: We introduce atoms for dyadic atomic \(\mathbb{H}^1\) for which the equivalence between the atomic and maximal function definitions is dimension independent. We give sharp, up to \(\log(d)\) factor, estimates for the \(\mathbb{H}^1\rightarrow L^1\) norm of the special maximal function.Maximal probability inequalities in vector latticeshttps://zbmath.org/1496.460022022-11-17T18:59:28.764376Z"Divandar, Mahin Sadat"https://zbmath.org/authors/?q=ai:divandar.mahin-sadat"Sadeghi, Ghadir"https://zbmath.org/authors/?q=ai:sadeghi.ghadirSummary: We generalize some maximal probability inequalities, proven for a class of random variables, to the measure-free setting of Riesz spaces. We prove generalizations of the Kolmogorov inequality, Hájek-Rényi inequality, Lévy's inequality and Etemadi's inequality.Limit laws for $R$-diagonal variables in a tracial probability spacehttps://zbmath.org/1496.460652022-11-17T18:59:28.764376Z"Zhou, Cong"https://zbmath.org/authors/?q=ai:zhou.congSummary: We study the weak convergence of sums of \(\ast\)-free, identically distributed tracial \(R\)-diagonal variables. The result parallels earlier results about free additive convolution on the real line. In particular, we determine under which conditions an infinitesimal array yields a sequence that converges to a given infinitely divisible tracial \(R\)-diagonal distribution. Thus, much of the work concerning sums of free (in the sense of Voiculescu) identically distributed positive random variables can be translated to the tracial \(R\)-diagonal context.Conditional expectation, entropy, and transport for convex Gibbs laws in free probabilityhttps://zbmath.org/1496.460672022-11-17T18:59:28.764376Z"Jekel, David"https://zbmath.org/authors/?q=ai:jekel.davidSummary: Let \((X_1,\dots ,X_m)\) be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential \(V\), and let \((S_1,\dots,S_m)\) be a free semicircular family. For \(k<m\), we show that conditional expectations and conditional non-microstates free entropy given \(X_1,\dots,X_k\) arise as the large \(N\) limit of the corresponding conditional expectations and entropy for the \(N\times N\) random matrix models associated to \(V\). Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism \(\mathrm{W}^*(X_1,\dots,X_m)\to\mathrm{W}^*(S_1,\dots,S_m)\) that maps \(\mathrm{W}^*(X_1,\dots,X_k)\) to \(\mathrm{W}^*(S_1,\dots,S_k)\) for each \(k=1,\dots,m\) and that also witnesses the Talagrand inequality for the law of \((X_1,\dots,X_m)\) relative to the law of \((S_1,\dots,S_m)\).Polynomial birth-death processes and the 2nd conjecture of Valenthttps://zbmath.org/1496.470572022-11-17T18:59:28.764376Z"Bochkov, Ivan"https://zbmath.org/authors/?q=ai:bochkov.ivanSummary: The conjecture of \textit{G. Valent} [ISNM, Int. Ser. Numer. Math. 131, 227--237 (1999; Zbl 0935.30025)] about the type of Jacobi matrices with polynomially growing weights is proved.Approximation and mean field control of systems of large populationshttps://zbmath.org/1496.490202022-11-17T18:59:28.764376Z"Higuera-Chan, Carmen G."https://zbmath.org/authors/?q=ai:higuera-chan.carmen-gSummary: We deal with a class of discrete-time stochastic controlled systems composed by a large population of \(N\) interacting individuals. Given that \(N\) is large and the cost function is possibly unbounded, the problem is studied by means of a limit model \(\mathcal{M}\), known as the mean field model, which is obtained as limit as \(N \rightarrow \infty\) of the model \(\mathcal{M}_N\) corresponding to the system of \(N\) individuals in combination with an approximate algorithm for the cost function.
For the entire collection see [Zbl 1478.60006].Isoperimetric clusters in homogeneous spaces via concentration compactnesshttps://zbmath.org/1496.490232022-11-17T18:59:28.764376Z"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Paolini, Emanuele"https://zbmath.org/authors/?q=ai:paolini.emanuele"Stepanov, Eugene"https://zbmath.org/authors/?q=ai:stepanov.eugene"Tortorelli, Vincenzo Maria"https://zbmath.org/authors/?q=ai:tortorelli.vincenzo-mariaSummary: We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural ``relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being minimal among all generalized clusters with the same masses. The examples include any anisotropic perimeter in a Euclidean space, as well as a hyperbolic plane with the Riemannian perimeter and Heisenberg groups with a canonical left invariant perimeter or its equivalent versions.On the Gaussian isoperimetric inequalityhttps://zbmath.org/1496.510112022-11-17T18:59:28.764376Z"Thomas, Erik"https://zbmath.org/authors/?q=ai:thomas.erik-g-fThe author first gives an elementary proof of an isoperimetric inequality for a Gaussian measure \(\gamma_1\) on \(\mathbb R\). Then as an application he proves one inequality of Pisier.
Reviewer: Martin Lukarevski (Skopje)Asymptotic geometric analysis. Part IIhttps://zbmath.org/1496.520012022-11-17T18:59:28.764376Z"Artstein-Avidan, Shiri"https://zbmath.org/authors/?q=ai:artstein-avidan.shiri"Giannopoulos, Apostolos"https://zbmath.org/authors/?q=ai:giannopoulos.apostolos-a"Milman, Vitali D."https://zbmath.org/authors/?q=ai:milman.vitali-dThis book is the continuation of the excellent monograph [\textit{S. Artstein-Avidan} et al., Asymptotic geometric analysis. I. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1337.52001)]. In this series of two books the modern theory of Asymptotic Geometric Analysis is presented. This theory had its origin in the field of (infinite dimensional) Functional Analysis, and evolved largely to a finite dimensional theory, but where the dimension of the objects of study (normed spaces, convex bodies, convex functions...) is very high, increasing to infinity. Both monographs are outstanding and essential references for the study of this theory.
The first four chapters are the natural continuation of the first volume. Thus, in Chapter 1 the authors revisit and deepen the concentration of measure phenomenon (cf. Chapter 3 of Part I), including its relation with important functional inequalities: First, Poincaré's inequality and its role in concentration is studied, and a proof of it in the Gaussian space based on the Ornstein-Uhlenbeck semigroup is provided; also the concentration on the discrete cube is exhaustively analyzed. Then, cost-induced transforms are discussed, including inequalities such as (Weak) Cost-Santaló inequality. I would like to mention another major inequality connected to this question of concentration, the logarithmic Sobolev inequality, which is also studied thoroughly.
Chapter 2 is devoted to discussing major results and problems on isotropic log-concave probability measures, being a follow-up of Chapter 10 in Part I. Among others, the authors deal with the famous Kannan-Lovász-Simonovits conjecture, its equivalent formulation saying that Poincaré's inequality holds for every isotropic log-concave probability measure on \(\mathbb{R}^n\) with a constant independent of the measure or the dimension, or the central limit problem. Important and breakthrough works on these questions by e.g. E. Milman, Eldan, Klartag or Chen are described.
The Gaussian distribution is one of the cornerstones in probability theory, and Chapter 3 deals with some fundamental isoperimetric inequalities about the \(n\)-dimensional Gaussian measure \(\gamma_n\) (cf. Chapter 9 of Part I). Naturally, one of the main results in this chapter is the engaging form of the isoperimetric inequality stating that if \(A\) is a Borel set in \(\mathbb{R}^n\) and \(H\) is a half-space with \(\gamma_n(A)=\gamma_n(H)\), then \(\gamma_n(A_t)\geq\gamma_n(H_t)\) for all \(t>0\), being \(A_t\) the \(t\)-extension of \(A\). Three different proofs of this important fact are presented (based on reducing it to the isoperimetric problem for the sphere, on a functional inequality or on a Gaussian symmetrization). Other significant results on this topic are studied here, such as Ehrhard's inequality, the behavior of the Gaussian measure with respect to dilates of a centrally symmetric convex body, the Gaussian correlation inequality, or the \(B\)-theorem of Cordero-Erausquin, Fradelizi and Maurey. Some applications of geometric inequalities for the Gaussian measure to discrepancy problems are also presented.
In Chapter 4, different volume-type inequalities are studied, continuing so the analysis that was made in Chapters 2 and 10 of Part I. First, the Brascamp-Lieb-Luttinger inequality, and the multidimensional versions of the Brascamp-Lieb and the Barthe inequalities are discussed, and several applications of these thorough results to different problems in Convex Geometry are presented. We highlight the Gluskin-Milman theorem, which is applied to show that every \(n\)-dimensional normed space has the random cotype-2 property. Next, volume estimates for convex bodies with few vertices or facets are exposed, as Vaaler's inequality bounding from below the volume of the intersection of a finite number of origin-symmetric strips, and other related results. Shephard's problem is also discussed, as well as its (strongly) negative answers by Petty and Schneider, and by Ball. Then the authors outline a theory that has been developed in several works of Paouris, Pivovarov et al., which provides a unified way of showing well-known inequalities from geometric probability. The chapter concludes considering the Blaschke-Petkantschin formulae and their geometric applications. Among them, Giannopoulos-Koldobsky's positive answer to a variant of the Busemann-Petty (and Shephard) problem, proposed by Milman, is presented.
Chapter 5 is devoted to the delightful theory of type and cotype, introduced and developed mainly by Maurey and Pisier. It starts with several basic notions and facts, and continues discussing the absolutely summing operators, nuclear operators, trace duality, the Gaussian type and cotype and the \(\ell\)-norm. Then, some developments on the duality of entropy problem for spaces with type \(p\) are analyzed, as well as some results for spaces with bounded cotype constant: the so-called ``Maurey-Pisier lemma'' and a theorem by Bourgain and Milman asserting that the volume ratio of the unit ball of an \(n\)-dimensional normed space is bounded by a function of the cotype 2 constant of the space. The last half of the chapter focuses on: Grothendieck's inequality; Kwapien's theorem asserting that the (Banach-Mazur) distance from a Banach space \(X\) to some Hilbert space is bounded from above by the product of the type 2 and the cotype 2 constants of \(X\); a theorem of Lindenstrauss and Tzafriri providing a positive answer to the complemented subspace problem; and Krivine's theorem and a counterpart/strengthening of it by Maurey and Pisier. A result of Johnson and Schechtman about embedding \(\ell^m_p\) into \(\ell^n_1\) closes the chapter.
Next, in Chapter 6 the geometry of the family of all normed (\(n\)-dimensional) spaces equipped with the Banach-Mazur distance (i.e., the so-called Banach-Mazur compactum) is investigated. The question of computing the diameter of the compactum is the starting point of the chapter, and Gluskin's theorem is shown: there exists an absolute constant \(c>0\) such that, for any \(n\in\mathbb{N}\), there exist two \(n\)-dimensional normed spaces with distance greater than \(cn\). Also the problem to estimate the diameter of the compactum in the non-symmetric case is discussed (the best known upper bound is due to Rudelson), for which the method of random orthogonal factorizations is previously introduced, as well as several applications of it. Next the authors consider Pełcynski's question about the asymptotic growth of the radius of the Banach-Mazur compactum with respect to \(\ell^n_{\infty}\); this problem is still open, the best known upper bound due to Giannopoulos being \(O(n^{5/6})\). The chapter ends with the study of several results from the local theory of normed spaces: Elton's theorem; a result by Milman-Wolfson about spaces with maximal distance to the Euclidean space; Alon-Milman's theorem on the existence of \(k\)-dimensional subespaces of a normed space with small distance to either \(\ell^k_2\) or \(\ell^k_{\infty}\); and the Schechtman theorem on the dependence on \(\varepsilon\) of the critical dimension in Dvoretzky theorem.
Symmetrizations of sets are one of the cornerstones in Convexity, and a major question in this matter is knowing how fast a sequence of symmetrizations approaches an Euclidean ball starting from an arbitrary convex body. Chapter 7 focuses on this issue, departing from two important symmetrizations: the Steiner and the Minkowski symmetrizations. While the first one has been most commonly studied in other monographs (although not from the point of view here considered), it is not the case for the second one, so conferring an extra incentive to the reading of this chapter. It is worthwhile to stress that in order to achieve this goal, the authors use tools from Asymptotic Geometric Analysis rather than the methods in classical Convex Geometry, which allow to show that the number of required symmetrizations in order to get ``close'' to an Euclidean ball is linear in the dimension. Works of Bourgain, Lindenstrauss and Milman, and mainly of Klartag, on this regard are presented, for both the Minkowski and the Steiner symmetrizations.
Next, Chapter 8 deals with the method of interlacing families of polynomials, focusing on its applications to Geometric Functional Analysis and Convex Geometry. It starts with a result of Batson, Spielman and Srivastava asserting, geometrically speaking, that a John decomposition of the identity can be approximated by a John sub-decomposition with suitable weights, involving a linear (in the dimension) number of terms. Then the interlacing polynomials are studied, and used afterwards for proving the restricted invertibility principle; other forms and generalizations of this theorem are also considered, one of which allows to show the proportional Dvoretzky-Rogers factorization theorem. The chapter ends with a full overview of the Kadison-Singer problem.
The book concludes with a wonderful Chapter 9, entitled ``Functionalization of Geometry'', where several sorts of functions on \(\mathbb{R}^n\) are studied from a geometric point of view: among them, of course, log-concave functions are the leading ones. Their importance in this context lies in the fact that they can be seen as an extension of (closed) convex sets, and particular operations between them as fundamental constructions in convex geometry; as an outcome, every geometric inequality has an analytic counterpart. Thus, after a first section devoted to log-concavity, functional duality is studied and, among others, it is proved that the Legendre transform is, up to linear variants, the only transform \(T\) defined on the set of convex functions on \(\mathbb{R}^n\) satisfying that \(T\circ T=Id\) and that \(\varphi\leq\psi\) if and only if \(T\varphi\geq T\psi\). The chapter ends showing functional versions of several fundamental geometric inequalities: Brunn-Minkowski, Uryshon, Blaschke-Santaló, reverse Brunn-Minkowski and Blaschke-Santaló, Rogers-Shephard, etc.
In line with the first volume of this series, all chapters are enriched with a final section, a collection of ``Notes and remarks'', where the main references regarding the content of the chapter can be found, as well as applications and many other problems related to its subject. The bibliography is large and exhaustive, consisting of almost 750 items, and including both, classical and very recent and seminal references.
This series of two books will most certainly be a fundamental source for the study and investigation in the field of Asymptotic Geometric Analysis.
Reviewer: Maria A. Hernández Cifre (Murcia)Symmetrized Talagrand inequalities on Euclidean spaceshttps://zbmath.org/1496.520132022-11-17T18:59:28.764376Z"Tsuji, Hiroshi"https://zbmath.org/authors/?q=ai:tsuji.hiroshi.1From the Introduction: The Talagrand inequality, which is also called the Talagrand transportation inequality, is as follows: If \(m = e^{-V}\, L_n\) is a probability measure on \(\mathbb{R}^n\) with \(\nabla^2V\ge k\) for some \(k>0\), and \(\mu\in P_2(\mathbb{R}^n)\), then \(W_2^2(\mu, m) \le \mathrm{Ent}_m(\mu)/k\) holds, where \(L_n\) is the Lebesgue measure on \(\mathbb{R}^n\), \(P_2(\mathbb{R}^n)\) is the set of all probability measures on \(\mathbb{R}^n\) with finite second moment, \(W_2\) is the Wasserstein distance, and \(\mathrm{Ent}_m\) is the relative entropy (or the Kullback-Leibler distance) with respect to \(m\). More generally, it is known that the Talagrand inequality holds on metric measure spaces with similar conditions to those above, and there are many studies on refinements of the Talagrand inequality and relations with logarithmic Sobolev inequalities and Poincaré inequalities [\textit{C. Villani}, Optimal transport. Old and new. Berlin: Springer (2009; Zbl 1156.53003)]. This paper is motivated by \textit{M. Fathi}'s following result [Electron. Commun. Probab. 23, Paper No. 81, 9 p. (2018; Zbl 1400.35008)]:
Theorem 1. Let \(\mu\), \(\nu\in P_2(\mathbb{R}^n)\). The following assertions hold:
\begin{itemize}
\item[(\textbf{1})] Let \(m = e^{-V}\, L_n\) be a probability measure on \(\mathbb{R}^n\) such that \(V\in C^{\infty}(\mathbb{R}^n)\) is even and \(k\)-convex for some \(k>0\). If \(\nu\) is symmetric (i.e., its density with respect to \(L_n\) is even), then it holds that \(1/2\, W_2^2(\mu, \nu) \le 1/k (\mathrm{Ent}_m(\mu) + \mathrm{Ent}_m(\nu))\).
\item[(\textbf{2})] Let \(m=\gamma_n\) be the \(n\)-dimensional standard Gaussian measure. If \(\int_{\mathbb{R}^n}x\, d\nu(x)=0\), then it holds that \(1/2\, W_2^2(\mu, \nu) \le \mathrm{Ent}_{\gamma_n}(\mu) + \mathrm{Ent}_{\gamma_n}(\nu))\).
\end{itemize}
Moreover, the equality holds in (\textbf{2}) if and only if there exist some positive definite symmetric matrix \(A\in \mathbb{R}^{n\times n}\) and some \(a\in \mathbb{R}\) such that \(\mu\) is the Gaussian measure whose center is \(a\) and covariance matrix is \(A\), and \(\nu\) is the Gaussian measure whose center is \(0\) and covariance matrix is \(A^{-1}\).
Note that Theorem 1 (\textbf{1}) does not include (\textbf{2}). When \(m=\nu\) in (\textbf{1}), we recover the classical Talagrand inequality, and hence Theorem 1 is a refinement of the classical Talagrand inequality provided \(V\) is even. Using Fathi's paper as reference, the author of the present paper calls this type of inequality a symmetrized Talagrand inequality. M. Fathi proved the symmetrized Talagrand inequality by using optimal transport theory and convex geometry. Moreover, M. Fathi pointed out that the symmetrized Talagrand inequality for Gaussian measures is related to the functional Blaschke-Santalo inequality, which is well known and important in convex geometry. This paper considers refinements and extensions of the symmetrized Talagrand inequality.
The present paper is organized as follows. Some fundamental notions from optimal transport theory and functional inequalities are introduced, including Talagrand inequalities. Section 3 gives another form of the symmetrized Talagrand inequality by a self-improvement of Fathi's result (Theorem 1 (\textbf{2})). The barycenter of a probability measure plays an important role in this section. In Section 4, the author proves the main theorems and apply them to prove the corresponding \(HWI\) inequalities (between entropy \(H\), Wasserstein distance \(W\) and Fisher information \(I\)), logarithmic Sobolev inequalities and Poincaré inequalities. Moreover, the author gives an alternative proof of Theorem 1 and an extension on the real line in Section 4.3. In the final section, the author describes an application of the result in the previous subsection to the Blaschke-Santalo inequality for log-concave probability measures.
Reviewer: Viktor Ohanyan (Erevan)Gaussian free fields and Riemannian rigidityhttps://zbmath.org/1496.530612022-11-17T18:59:28.764376Z"Nguyen Viet Dang"https://zbmath.org/authors/?q=ai:nguyen-viet-dang.Summary: On a compact Riemannian manifold \((M,g)\) of dimension \(d\leqslant 4\), we present a rigorous construction of the renormalized partition function \(Z_g(\lambda)\) of a massive Gaussian free field where we explicitly determine the local counterterms using microlocal methods. Then we show that \(Z_g(\lambda)\) determines the Laplace spectrum of \((M,g)\) and hence imposes some strong geometric constraints on the Riemannian structure of \((M,g)\). From this observation, using classical results in Riemannian geometry, we illustrate how the partition function allows us to probe the Riemannian structure of the underlying manifold \((M,g)\).Elliptic gradient estimates for a nonlinear \(f\)-heat equation on weighted manifolds with evolving metrics and potentialshttps://zbmath.org/1496.531002022-11-17T18:59:28.764376Z"Abolarinwa, Abimbola"https://zbmath.org/authors/?q=ai:abolarinwa.abimbola"Taheri, Ali"https://zbmath.org/authors/?q=ai:taheri.ali|taheri.ali-karimiSummary: We develop local elliptic gradient estimates for a basic nonlinear \(f\)-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed.Soft \(\beta\)-rough sets and their application to determine COVID-19https://zbmath.org/1496.540012022-11-17T18:59:28.764376Z"El Bably, Mostafa K."https://zbmath.org/authors/?q=ai:el-bably.mostafa-k"El Atik, Abd El Fattah A."https://zbmath.org/authors/?q=ai:el-atik.abd-el-fattah-aSummary: Soft rough set theory has been presented as a basic mathematical model for decision-making for many real-life data. However, soft rough sets are based on a possible fusion of rough sets and soft sets which were proposed by \textit{F. Feng} et al. [Inf. Sci. 181, No. 6, 1125--1137 (2011; Zbl 1211.68436)]. The main contribution of the present article is to introduce a modification and a generalization for Feng's approximations, namely, soft \(\beta\)-rough approximations, and some of their properties will be studied. A comparison between the suggested approximations and the previous one [loc. cit.] will be discussed. Some examples are prepared to display the validness of these proposals. Finally, we put an actual example of the infections of coronavirus (COVID-19) based on soft \(\beta\)-rough sets. This application aims to know the persons most likely to be infected with COVID-19 via soft \(\beta\)-rough approximations and soft \(\beta\)-rough topologies.Riesz probability distributionshttps://zbmath.org/1496.600012022-11-17T18:59:28.764376Z"Hassairi, Abdelhamid"https://zbmath.org/authors/?q=ai:hassairi.abdelhamidFrom the cover of the book:\\
``Unique in the literature, this book provides an introductory, comprehensive and essentially self-contained exposition of the Riesz probability distribution on a symmetric cone and of its derivatives, with an emphasis on the case of the cone of positive definite symmetric matrices. \\
This distribution is an important generalization of the Wishart whose definition relies on the notion of generalized power. \\
Researchers in probability theory and harmonic analysis will find this book to be an important resource. \\
Given the connection between the Riesz probability distribution and the multivariate Gaussian samples with missing data, the book is also accessible and useful for statisticians.'' \\
\\
The preface finishes with:\\
``I hope that the book will be useful as a source of statements and applications of results in multivariate probability distributions and multivariate statistical analysis, as well as a reference to some material of harmonic analysis on symmetric cones adapted to the needs of researchers in these fields.'' \\
\\
The book is very large structured in Contents, Preface, Acknowledgment, 11 Chapters (with 45 subchapters), Bibliography (with 117 references), Index (with more than 70 items), Index of notations: \\
Chapter 1. Jordan algebras and symmetric cones -- Chapter 2. Generalized power -- Chapter 3. Riesz probability distributions -- Chapter 4. Riesz natural exponential families -- Chapter 5. Tweedie scale -- Chapter 6. Moments and constancy of regression -- Chapter 7. Beta Riesz probability distributions -- Chapter 8. Beta-Wishart distributions -- Chapter 9. Beta-hypergeometric distributions -- Chapter 10. Riesz-Dirichlet distributions -- Chapter 11. Riesz inverse Gaussian distribution \\
\\
The book can be very recommended all readers who are interested in this field.
Reviewer: Ludwig Paditz (Dresden)Probability and random variables: theory and applicationshttps://zbmath.org/1496.600022022-11-17T18:59:28.764376Z"Song, Iickho"https://zbmath.org/authors/?q=ai:song.iickho"Park, So Ryoung"https://zbmath.org/authors/?q=ai:park.so-ryoung"Yoon, Seokho"https://zbmath.org/authors/?q=ai:yoon.seokhoPublisher's description: This book discusses diverse concepts and notions -- and their applications -- concerning probability and random variables at the intermediate to advanced level. It explains basic concepts and results in a clearer and more complete manner than the extant literature. In addition to a range of concepts and notions concerning probability and random variables, the coverage includes a number of key advanced concepts in mathematics. Readers will also find unique results on e.g. the explicit general formula of joint moments and the expected values of nonlinear functions for normal random vectors. In addition, interesting applications of the step and impulse functions in discussions on random vectors are presented. Thanks to a wealth of examples and a total of 330 practice problems of varying difficulty, readers will have the opportunity to significantly expand their knowledge and skills. The book is rounded out by an extensive index, allowing readers to quickly and easily find what they are looking for.
Given its scope, the book will appeal to all readers with a basic grasp of probability and random variables who are looking to go one step further. It also offers a valuable reference guide for experienced scholars and professionals, helping them review and refine their expertise.Correction to: ``On the equivalence of conglomerability and disintegrability for unbounded random variables''https://zbmath.org/1496.600032022-11-17T18:59:28.764376Z"Schervish, Mark J."https://zbmath.org/authors/?q=ai:schervish.mark-j"Seidenfeld, Teddy"https://zbmath.org/authors/?q=ai:seidenfeld.teddy"Kadane, Joseph B."https://zbmath.org/authors/?q=ai:kadane.joseph-bornCorrection to the authors' paper [ibid. 23, No. 4, 501--518 (2014; Zbl 1477.60011)].A method of defining central and Gibbs measures and the ergodic methodhttps://zbmath.org/1496.600042022-11-17T18:59:28.764376Z"Vershik, A. M."https://zbmath.org/authors/?q=ai:vershik.anatoli-mAuthor's abstract: We formulate a general statement of the problem of defining invariant measures with certain properties and suggest an ergodic method of perturbations for describing such measures.
Reviewer: Göran Högnäs (Åbo)Convergence rates for empirical measures of Markov chains in dual and Wasserstein distanceshttps://zbmath.org/1496.600052022-11-17T18:59:28.764376Z"Riekert, Adrian"https://zbmath.org/authors/?q=ai:riekert.adrianSummary: We consider a Markov chain on \(\mathbb{R}^d\) with invariant measure \(\mu \). We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the 1-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.Proof of Tomaszewski's conjecture on randomly signed sumshttps://zbmath.org/1496.600062022-11-17T18:59:28.764376Z"Keller, Nathan"https://zbmath.org/authors/?q=ai:keller.nathan"Klein, Ohad"https://zbmath.org/authors/?q=ai:klein.ohadLet \(\{x_i\}\) be as sequence of independent and uniformly distributed in \(\{-1, 1\}\) random variables. Consider the sum
\[
X =\sum_{i=1}^n a_ix_i,
\]
where \( \{a_i\}\) are real numbers such that \(\sum_{i=1}^n a_i^2= 1.\) The authors prove Tomaszewski's conjecture:
\[
P\left[|X| \le 1\right] \ge 1/2.
\]
The main tools to obtain the result are a new local concentration inequality, an improved Berry-Esseen inequality for Rademacher sums and a stopping time argument. The paper concludes with several open problems.
Reviewer: Andriy Olenko (Melbourne)The fluid limit of a random graph model for a shared ledgerhttps://zbmath.org/1496.600072022-11-17T18:59:28.764376Z"King, Christopher"https://zbmath.org/authors/?q=ai:king.christopher-kSummary: A shared ledger is a record of transactions that can be updated by any member of a group of users. The notion of independent and consistent record-keeping in a shared ledger is important for blockchain and more generally for distributed ledger technologies. In this paper we analyze a stochastic model for the shared ledger known as the tangle, which was devised as the basis for the IOTA cryptocurrency. The model is a random directed acyclic graph, and its growth is described by a non-Markovian stochastic process. We first prove ergodicity of the stochastic process, and then derive a delay differential equation for the fluid model which describes the tangle at high arrival rate. We prove convergence in probability of the tangle process to the fluid model, and also prove global stability of the fluid model. The convergence proof relies on martingale techniques.Two-degree-of-freedom Ellsberg urn problemhttps://zbmath.org/1496.600082022-11-17T18:59:28.764376Z"Lio, Waichon"https://zbmath.org/authors/?q=ai:lio.waichon"Cheng, Guangquan"https://zbmath.org/authors/?q=ai:cheng.guangquanSummary: Traditional model assumes there is only randomness existing in the urn problem. However, if the numbers of the colored balls are unknown, then they should be regarded as uncertain variable. Since a ball is drawn randomly, Ellsberg urn problem is essentially a complicated system with randomness and uncertainty. Instead of psychological experiment, this paper applies uncertainty theory and chance theory to provide a rigorous mathematical method for formulating the general case of a one-degree-of-freedom Ellsberg urn problem. Furthermore, a two-degree-of-freedom Ellsberg urn problem is proposed, and the formulation for the problem is given to deal with three unknown numbers of colored balls.First passage times of subordinators and urnshttps://zbmath.org/1496.600092022-11-17T18:59:28.764376Z"Marchal, Philippe"https://zbmath.org/authors/?q=ai:marchal.philippeSummary: It is well-known that the first time a stable subordinator reaches \([1, +\infty)\). is Mittag-Leffler distributed. These distributions also appear as limiting distributions in triangular Polya urns. We give a direct link between these two results, using a previous construction of the range of stable subordinators. Beyond the stable case, we show that for a subclass of complete subordinators in the domain of attraction of stable subordinators, the law of the first passage time is given by the limit of an urn with the same replacement rule but with a random initial composition.
For the entire collection see [Zbl 1478.60005].Mosco convergence of strong laws of large numbers for triangular array of row-wise exchangeable random sets and fuzzy random setshttps://zbmath.org/1496.600102022-11-17T18:59:28.764376Z"Nguyen Van Quang"https://zbmath.org/authors/?q=ai:nguyen-van-quang."Duong Xuan Giap"https://zbmath.org/authors/?q=ai:duong-xuan-giap.Summary: In this paper, we obtain strong laws of large numbers for triangular array of row-wise exchangeable random sets and fuzzy random sets in a separable Banach space in the Mosco sense. Our results are obtained without bounded expectation condition, with or without compactly uniformly integrable and reverse martingale hypotheses. They improve some related results in literature. Moreover, some typical examples illustrating this study are provided.Truncated Mittag-Leffler distribution and superstatisticshttps://zbmath.org/1496.600112022-11-17T18:59:28.764376Z"Agahi, Hamzeh"https://zbmath.org/authors/?q=ai:agahi.hamzeh"Khalili, Monavar"https://zbmath.org/authors/?q=ai:khalili.monavarSummary: The Mittag-Leffler stochastic process is an important tool in practical applications. In this paper, we first focus on some claims of Burr type-XII as a superstatistical stationary distribution
[\textit{E. Sánchez}, Physica A 516, 443--446 (2019; Zbl 07558193)]. Then, we present the capabilities of the Mittag-Leffler distribution and introduce truncated Mittag-Leffler distributions. Finally, in the oil price analysis of time series real data, we show that the truncated Mittag-Leffler distribution performs better than other nominated distributions, especially the Burr distribution.Roots of random functions: a framework for local universalityhttps://zbmath.org/1496.600122022-11-17T18:59:28.764376Z"Nguyen, Oanh"https://zbmath.org/authors/?q=ai:nguyen.oanh"Vu, Van"https://zbmath.org/authors/?q=ai:vu.van-hLet \(\phi_0(z),\ldots, \phi_n(z)\) be deterministic analytic functions and \(\xi_0,\ldots,\xi_n\) independent random variables. The authors study the distribution of zeroes (both, real and complex) of the random functions of the form \[ F_n(z) = \sum_{k=0}^n \xi_k \phi_k(z). \] This general setting includes many special cases such as \(\phi_k(z) = z^k\) corresponding to the so-called Kac polynomials, \(\phi_k(z) = \cos (k z)\) (corresponding to random trigonometric polynomials) or, more generally, \(\phi_k(z) = c_k z^k\) or \(\phi_k(z) = c_k \cos (k z)\) with deterministic \(c_k\)'s satisfying appropriate assumptions. The authors develop a general framework to study the local distribution of roots of such functions via ``universality theorems'' which state that the roots of \(F_n\) for general \(\xi_k\)'s can be approximated by the roots of the function \(\tilde F_n\) in which the \(\xi_k\)'s are Gaussian. More precisely, they prove explicit upper error bounds on differences of the form \[ \left|\mathbb E \sum_{i_1,\ldots, i_k} G(\zeta_{i_1},\ldots, \zeta_{i_k}) - \mathbb E \sum_{i_1,\ldots, i_k} G(\tilde \zeta_{i_1},\ldots, \tilde \zeta_{i_k})\right|, \] where \(G\) is a sufficiently smooth function. Here, the first sum runs over all \(k\)-tuples of roots of \(F_n\), while the second sum runs over all \(k\)-tuples of the roots of \(\tilde F_n\) in which the \(\xi_k\)'s are replaced by Gaussian random variables. As an application of their techniques, the authors recover, extend in a unified way and sharpen many known results on the asymptotics of the expected number of real roots of random analytic functions including such examples as Kac, Weyl and elliptic random polynomials, random trigonometric polynomials and random Taylor series with regularly varying coefficients.
Reviewer: Zakhar Kabluchko (Münster)On multivariate quasi-infinitely divisible distributionshttps://zbmath.org/1496.600132022-11-17T18:59:28.764376Z"Berger, David"https://zbmath.org/authors/?q=ai:berger.david"Kutlu, Merve"https://zbmath.org/authors/?q=ai:kutlu.merve"Lindner, Alexander"https://zbmath.org/authors/?q=ai:lindner.alexander-mSummary: A quasi-infinitely divisible distribution on \(\mathbb{R}^d\) is a probability distribution \(\mu\) on \(\mathbb{R}^d\) whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on \(\mathbb{R}^d\). Equivalently, it can be characterised as a probability distribution whose characteristic function has a Lévy-Khintchine type representation with a ``signed Lévy measure'', a so called quasi-Lévy measure, rather than a Lévy measure. A systematic study of such distributions in the univariate case has been carried out in [the last author et al., Trans. Am. Math. Soc. 370, No. 12, 8483--8520 (2018; Zbl 1428.60034)]. The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on \(\mathbb{Z}^d\)-valued quasi-infinitely divisible distributions.
For the entire collection see [Zbl 1478.60005].Smoothing effect and derivative formulas for Ornstein-Uhlenbeck processes driven by subordinated cylindrical Brownian noiseshttps://zbmath.org/1496.600142022-11-17T18:59:28.764376Z"Bondi, Alessandro"https://zbmath.org/authors/?q=ai:bondi.alessandroSummary: We investigate the concept of cylindrical Wiener process subordinated to a strictly \(\alpha\)-stable Lévy process, with \(\alpha \in (0, 1)\), in an infinite-dimensional, separable Hilbert space, and consider the related stochastic convolution. We then introduce the corresponding Ornstein-Uhlenbeck process, focusing on the regularizing properties of the Markov transition semigroup defined by it. In particular, we provide an explicit, original formula -- which is not of Bismut-Elworthy-Li's type -- for the Gateaux derivatives of the functions generated by the operators of the semigroup, as well as an upper bound for the norm of their gradients. In the case \(\alpha \in (\frac{1}{2}, 1)\), this estimate represents the starting point for studying the Kolmogorov equation in its mild formulation.Some new classes and techniques in the theory of Bernstein functionshttps://zbmath.org/1496.600152022-11-17T18:59:28.764376Z"Bridaa, Safa"https://zbmath.org/authors/?q=ai:bridaa.safa"Fourati, Sonia"https://zbmath.org/authors/?q=ai:fourati.sonia"Jedidi, Wissem"https://zbmath.org/authors/?q=ai:jedidi.wissemSummary: In this paper we provide some new properties that are complementary to the book of \textit{R. L. Schilling} et al. [Bernstein functions. Theory and applications. 2nd revised and extended ed. Berlin: de Gruyter (2012; Zbl 1257.33001)].
For the entire collection see [Zbl 1478.60005].Correction to: ``Formulas of absolute moments''https://zbmath.org/1496.600162022-11-17T18:59:28.764376Z"Lin, Gwo Dong"https://zbmath.org/authors/?q=ai:lin.gwo-dong"Hu, Chin-Yuan"https://zbmath.org/authors/?q=ai:hu.chinyuanCorrection to the authors' paper [ibid. 83, No. 1, 476--495 (2021; Zbl 1459.60042)].Stochastic comparison for elliptically contoured random fieldshttps://zbmath.org/1496.600172022-11-17T18:59:28.764376Z"Lu, Tianshi"https://zbmath.org/authors/?q=ai:lu.tianshi"Du, Juan"https://zbmath.org/authors/?q=ai:du.juan"Ma, Chunsheng"https://zbmath.org/authors/?q=ai:ma.chunshengSummary: This paper presents necessary and sufficient conditions for the peakedness comparison and convex ordering between two elliptically contoured random fields about their centers. A somewhat surprising finding is that the peakedness comparison for the infinite dimensional case differs from the finite dimensional case. For example, a Student's t distribution is known to be more heavy-tailed than a normal distribution, but a Student's t random field and a Gaussian random field are not comparable in terms of the peakedness. In particular, the peakedness comparison and convex ordering are made for isotropic elliptically contoured random fields on compact two-point homogeneous spaces.Central limit theorem for the antithetic multilevel Monte Carlo methodhttps://zbmath.org/1496.600182022-11-17T18:59:28.764376Z"Alaya, Mohamed Ben"https://zbmath.org/authors/?q=ai:alaya.mohamed-ben"Kebaier, Ahmed"https://zbmath.org/authors/?q=ai:kebaier.ahmed"Ngo, Thi Bao Tram"https://zbmath.org/authors/?q=ai:ngo.thi-bao-tramSummary: In this paper, we give a natural extension of the antithetic multilevel Monte Carlo (MLMC) estimator for a multidimensional diffusion introduced by \textit{M. B. Giles} and \textit{L. Szpruch} [Ann. Appl. Probab. 24, No. 4, 1585--1620 (2014; Zbl 1373.65007)] by considering the permutation between \(m\) Brownian increments, \(m\geq 2\), instead of using two increments as in the original paper. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its \(\sigma\)-antithetic version build on the finer grid, and on the other hand, the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its \(\sigma\)-antithetic version stably converges in distribution with a rate of order \(1/2\). More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see, e.g., [\textit{J. Jacod}, Lect. Notes Math. 1655, 232--246 (1997; Zbl 0884.60038)]). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the antithetic MLMC estimator. The time complexity of the algorithm is analyzed.The limit distribution of a singular sequence of Itô integralshttps://zbmath.org/1496.600192022-11-17T18:59:28.764376Z"Bell, Denis"https://zbmath.org/authors/?q=ai:bell.denis-rSummary: We give an alternative, elementary proof of a result of \textit{G. Pecati} and \textit{M. Yor} [Fields Inst. Commun. 44, 75--87 (2004; Zbl 1071.60017)] concerning the limit law of a sequence of Itô integrals with integrands having singular asymptotic behavior.
For the entire collection see [Zbl 1478.60005].The elephant random walk with gradually increasing memoryhttps://zbmath.org/1496.600202022-11-17T18:59:28.764376Z"Gut, Allan"https://zbmath.org/authors/?q=ai:gut.allan"Stadtmüller, Ulrich"https://zbmath.org/authors/?q=ai:stadtmuller.ulrichThe elephant random walk is given by \(S_n=X_1+\cdots+X_n\), where \(X_1=1\) with probability \(p\) and \(X_1=-1\) with probability \(1-p\). Subsequent steps are then chosen such that, for \(n\geq0\), \(X_{n+1}=X_K\) with probability \(p\) and \(X_{n+1}=-X_K\) with probability \(1-p\), where \(K\) is uniformly distributed over some subset \(\mathfrak{M}\) of \(\{1,2,\dots,n\}\). In the present paper, the authors mainly study the case where \(\mathfrak{M}=\{1,2,\dots,m_n\}\), where \(m_n\to\infty\) as \(n\to\infty\) in such a way that \(m_n/n\to0\). The authors establish asymptotics of the mean and variance of \(S_n\), and limit theorems for (suitably scaled versions of) \(S_n\). These asymptotics and limiting distributions depend on how \(p\) compares to the transitional value \(p=3/4\).
Asymptotics for moments and limit theorems are also derived for the case where the random walk has some positive probability of not moving at each step, and for the case where \(\mathfrak{M}=\{1,2,\dots,m_n,n\}\), i.e., the random walk also remembers the most recent step. Finally, asymptotics of the first two moments are also given for the case where \(m_n/n\to\alpha\in(0,1]\) as \(n\to\infty\). The question of limit theorems in this case is left open; in the previous cases the proofs of the limit theorems relied on certain conditional variances vanishing asymptotically, a property which no longer holds in this case.
Reviewer: Fraser Daly (Edinburgh)Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equationhttps://zbmath.org/1496.600212022-11-17T18:59:28.764376Z"Mukherjee, Chiranjib"https://zbmath.org/authors/?q=ai:mukherjee.chiranjibSummary: We consider a class of Gibbs measures defined with respect to increments \(\{\omega (t)-\omega (s)\}_{s<t}\) of \(d\)-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form \(H(t-s,\omega (t)-\omega (s))\) that are invariant under uniform translations of paths. In such interactions, we allow \textit{long-range} dependence in the time variable (including power law decay up to \(t\mapsto (1+t)^{-(2+\varepsilon)}\) for \(\varepsilon > 0)\) and \textit{unbounded (singular)} interactions (including singularities of the form \(x\mapsto 1/|x|^p\) in \(d\geq 3\) or \(x\mapsto \delta_0 (x)\) in \(d=1)\) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Fröhlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance, which is strictly positive.
As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions \(d\geq 3\). When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.On the weak laws of large numbers for compound random sums of independent random variables with convergence rateshttps://zbmath.org/1496.600222022-11-17T18:59:28.764376Z"Tran, Loc Hung"https://zbmath.org/authors/?q=ai:hung.tran-locSummary: The weak laws of large numbers are extremely intuitive and applicable results in various fields of probability theory and mathematical statistics. Compound random sums are extensions of classical random sums where the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, assuming independence of summands. In this paper, a weak laws of large numbers for normalized compound random sums of independent (not necessarily identically distributed) random variables is studied and the convergence rates in types of ``Small-o'' and ``Large-\(\mathcal{O}\)'' error estimates are established, using Trotter's distance approach. The obtained results in this paper are extensions and generalizations of the known classical ones.Large deviations and Wschebor's theoremshttps://zbmath.org/1496.600232022-11-17T18:59:28.764376Z"León, José R."https://zbmath.org/authors/?q=ai:leon.jose-rafael-r"Rouault, Alain"https://zbmath.org/authors/?q=ai:rouault.alainSummary: We revisit Wschebor's theorems on the a.s. convergence of small increments for processes with scaling and stationarity properties. We focus on occupation measures and proved that they satisfy large deviation principles.
For the entire collection see [Zbl 1478.60006].Deviations for weak record numbers in simple random walkshttps://zbmath.org/1496.600242022-11-17T18:59:28.764376Z"Li, Yuqiang"https://zbmath.org/authors/?q=ai:li.yuqiang"Yao, Qiang"https://zbmath.org/authors/?q=ai:yao.qiangThe main goal in this manuscript is to provide the asymptotic probabilities of large and moderate deviations for the number of weak records in one-dimensional symmetric simple random walks. Due to its widespread applicability, the study of record statistics plays an integral role in diverse fields, including but not limited to sports, hydrology, economics and others.
The main results are contained in two main theorems: Theorem 1 and Theorem 3. Applications of these results are also presented.
Reviewer: Maria C. Mariani (El Paso)Large deviations of mean-field interacting particle systems in a fast varying environmenthttps://zbmath.org/1496.600252022-11-17T18:59:28.764376Z"Yasodharan, Sarath"https://zbmath.org/authors/?q=ai:yasodharan.sarath"Sundaresan, Rajesh"https://zbmath.org/authors/?q=ai:sundaresan.rajeshSummary: This paper studies large deviations of a ``fully coupled'' finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applicationshttps://zbmath.org/1496.600262022-11-17T18:59:28.764376Z"Bao, Xiaohan"https://zbmath.org/authors/?q=ai:bao.xiaohan"Lin, Junjie"https://zbmath.org/authors/?q=ai:lin.junjie"Wang, Xuejun"https://zbmath.org/authors/?q=ai:wang.xuejun"Wu, Yi"https://zbmath.org/authors/?q=ai:wu.yiSummary: In this paper, the complete convergence for the weighted sums of arrays of rowwise extended negatively dependent (END, for short) random variables is established under some mild conditions. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for arrays of rowwise END random variables is also obtained. The result obtained in the paper generalizes and improves some corresponding ones for independent random variables and some dependent random variables in some extent. By using the complete convergence that we established, we further study the complete consistency for the weighted estimator in a nonparametric regression model based on END errors.A note on the cluster set of the law of the iterated logarithm under sub-linear expectationshttps://zbmath.org/1496.600272022-11-17T18:59:28.764376Z"Zhang, Li-Xin"https://zbmath.org/authors/?q=ai:zhang.li-xin|zhang.lixinThe main result of the present paper is a law of the iterated logarithm for a sequence of independent and identically distributed random variables in a sub-linear expectation space, with upper capacity. The proof proceeds via a self-normalized law of the iterated logarithm and versions of a Borel-Cantelli lemma for this setting.
Reviewer: Fraser Daly (Edinburgh)On simulating the short and long memory of ergodic Markov and non-Markov genetic diffusion processes on the long runhttps://zbmath.org/1496.600282022-11-17T18:59:28.764376Z"Abdel-Rehim, E. A."https://zbmath.org/authors/?q=ai:abdel-rehim.enstar-a|abdel-rehim.entsar-a"Hassan, R. M."https://zbmath.org/authors/?q=ai:hassan.r-m"El-Sayed, A. M. A."https://zbmath.org/authors/?q=ai:el-sayed.ahmed-mohamed-ahmedSummary: The theory of classical ergodic Markov genetic diffusion process, Feller genetic diffusion, and the genetic drift diffusion with and without selection and mutation for a single unlinked locus with two alleles, Kimura models, is studied. These genetic problems are numerically investigated on the long run. The approximates solutions of these models are described as the conditional probabilities to find the specific gene with frequencies between zero and one at generation \(T\). As a surprise, we notice that the summation of these approximate solutions lose their unity rapidly as the number of generations increase than five, i.e. \(T>>5\). The earlier biologists did due the reason to the migration, immigration and death of individuals. Till now, there is no mathematical proof to this phenomenon. In this paper, we solve this problem by extending the first order time derivative to the time fractional derivative, to study the effect of the memory on these models. This extension ensures that the summations of the approximate solutions, i.e. of the Non-Markov cases, are one at any number of generations. The time evolution of the approximate solutions are simulated and compared with other published papers, for different values of generations and for the Markov and Non-Markov cases with different values of the fractional order \(\beta\). The convergence of the approximate descrete solutions and the reversibility property of these stochastic processes for both the Markov cases and Non-Markov cases are also numerically simulated and discussed.Copula measures and Sklar's theorem in arbitrary dimensionshttps://zbmath.org/1496.600292022-11-17T18:59:28.764376Z"Benth, Fred Espen"https://zbmath.org/authors/?q=ai:benth.fred-espen"Di Nunno, Giulia"https://zbmath.org/authors/?q=ai:di-nunno.giulia"Schroers, Dennis"https://zbmath.org/authors/?q=ai:schroers.dennisSummary: Although copulas are used and defined for various infinite-dimensional objects (e.g., Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified functional analytic framework, show how Sklar's theorem can be applied in certain examples of Banach spaces and provide a semiparametric estimation procedure for second-order stochastic processes with underlying Gaussian copula.New generalized mean square stochastic fractional operators with applicationshttps://zbmath.org/1496.600302022-11-17T18:59:28.764376Z"Khan, Tahir Ullah"https://zbmath.org/authors/?q=ai:khan.tahir-ullah"Khan, Muhammad Adil"https://zbmath.org/authors/?q=ai:khan.muhammad-adilSummary: This paper aims to develop novel generalized mean square fractional integral and derivative operators (for second-order stochastic processes) and presents an application of these operators. To conduct our study, first, the Riemann-Liouville's approach is applied to iterate and fractionalize two specially constructed mean square Riemann integral operators. This procedure leads us to define new generalized left-sided and right-sided mean square fractional integral operators based on which the associated generalized mean square fractional derivative operators (left- and right-sided) are also defined. Some properties such as linearity, semigroup property, boundedness, continuity, and inverse property are investigated for these operators. It is proved that new classes of these left- and right-sided integral and derivative operators constitute Banach spaces and they also exist in the sense of probability. These operators generalize various mean square operators of the types Katugampola, Hadamard, Riemann-Liouville, Riemann, in which the operators of Riemann-Liouville and that of the Riemann have already been previously defined and the rest are new in the sense of mean square stochastic calculus. As an application, the newly established results are applied to prove a generalized fractional stochastic version of the well-known Hermite-Hadamard inequality for convex stochastic processes, where a bound for the absolute of the difference between the two rightmost terms is also obtained. Some relations of our results with the already existing results are also discussed.Transition path properties for one-dimensional systems driven by Poisson white noisehttps://zbmath.org/1496.600312022-11-17T18:59:28.764376Z"Li, Hua"https://zbmath.org/authors/?q=ai:li.hua"Xu, Yong"https://zbmath.org/authors/?q=ai:xu.yong.1"Metzler, Ralf"https://zbmath.org/authors/?q=ai:metzler.ralf"Kurths, Jürgen"https://zbmath.org/authors/?q=ai:kurths.jurgenSummary: We present an analytically tractable scheme to solve the mean transition path shape and mean transition path time of one-dimensional stochastic systems driven by Poisson white noise. We obtain the Fokker-Planck operator satisfied by the mean transition path shape. Based on the non-Gaussian property of Poisson white noise, a perturbation technique is introduced to solve the associated Fokker-Planck equation. Moreover, the mean transition path time is derived from the mean transition path shape. We illustrate our approximative theoretical approach with the three paradigmatic potential functions: linear, harmonic ramp, and inverted parabolic potential. Finally, the Forward Fluxing Sampling scheme is applied to numerically verify our approximate theoretical results. We quantify how the Poisson white noise parameters and the potential function affect the symmetry of the mean transition path shape and the mean transition path time.Selected topics in the generalized mixed set-indexed fractional Brownian motionhttps://zbmath.org/1496.600322022-11-17T18:59:28.764376Z"Yosef, Arthur"https://zbmath.org/authors/?q=ai:yosef.arthurLet \(\mathcal A\) be a compact set collection on a topological space \((T,\tau).\) A set-indexed fractional Brownian motion (sifbm) of Hurst parameter \(H\in (0, \frac{1}{2}]\) is a centered Gaussian process \(B^H=\{B_A^H:A \in \mathcal A\}\) defined on a probability space \((\Omega, \mathcal F,\mu)\) with the covariance function \[E[B_A^HB_B^H]= \frac{1}{2}[\mu(A)^{2H}+\mu(B)^{2H}-\mu(A\Delta B)^{2H}]\] for all \(A,B\in \mathcal A.\) The author studies properties of a generalized set-indexed fractional Brownian motion (gsifBm). The gsifBm of parameters \(H=\{H_1,\dots,H_n\}, \alpha= \{\alpha_1,\dots,\alpha_n\}, (0,H_i\leq \frac{1}{2}\) and \(\{ \alpha_i \}\) are real not all zero) is the process \[M_A^{H,\alpha}=\sum_{i=1}^n\alpha_iB_A^{H_i}\] for all \(A\in \mathcal A,\) where \(B^{H_i}\) are independent sifBm of parameter \(H_i, 1\leq i \leq n.\) This is a generalized mixed fractional Brownian motion in the set-indexed framework. The author investigates the properties of self-similarity, long-range dependence, Holder continuity, differentiability and Hausdorff dimension for such processes.
Reviewer: B. L. S. Prakasa Rao (Hyderabad)Corrigendum to: ``On iterated function systems with place-dependent probabilities''https://zbmath.org/1496.600332022-11-17T18:59:28.764376Z"Bárány, Balázs"https://zbmath.org/authors/?q=ai:barany.balazsFrom the text: In my paper [ibid. 143, No. 1, 419--432 (2015; Zbl 1308.60044)], the proof of the main theorem contains a crucial error. The estimates in page 427 line 2--5 and in page 429 line 6--7 are incorrect. The solution of this problem was far from being trivial and so the proof of this paper remains incomplete.
In our recent paper with my coauthors, \textit{K. Simon}, \textit{B. Solomyak} and \textit{A. Śpiewak}, we were able to circumvent the problem. Under more restrictive assumptions on the smoothness of a one-dimensional parameter family of iterated function systems, we showed that the claim of the main theorem remains valid even for a wider class of parameter dependent family of invariant measures. For details, we refer the reader to [Adv. Math. 399, Article ID 108258, 73 p. (2022; Zbl 07496420)].On optimal threshold stopping times for Ito diffusionshttps://zbmath.org/1496.600342022-11-17T18:59:28.764376Z"Arkin, V. I."https://zbmath.org/authors/?q=ai:arkin.vadim-i"Slastnikov, A. D."https://zbmath.org/authors/?q=ai:slastnikov.alexander-dSummary: The paper deals with an optimal stopping problem for one-dimensional Ito diffusion process and terminal payoff function. We study the following problem: under what conditions the stopping time which is optimal over the class of threshold strategies (specifying by first time when the underlying process exceeds some level) remains optimal over all stopping times. We prove that excessiveness of payoff function is the only necessary and sufficient condition which connects optimality over the class of threshold stopping times and over all stopping times.On the dimension reduction in the quickest detection problem for diffusion processes with exponential penalty for the delayhttps://zbmath.org/1496.600352022-11-17T18:59:28.764376Z"Buonaguidi, Bruno"https://zbmath.org/authors/?q=ai:buonaguidi.brunoThe author considers the problem of quickest detection of a change in the drift rate of an observable time-homogeneous diffusion process under the assumption that the detection delay is exponentially penalized. According to the existing literature, such problems are normally embedded into the corresponding optimal stopping problems for two-dimensional or three-dimensional Markov processes. In the paper under review, the author shows how a change of measure may significantly simplify the setting by means of reducing the appropriate dimension of the optimal stopping problem to one or two, respectively. The results are illustrated on the well-known cases of the observable Brownian motion analyzed by \textit{M. Beibel} [Ann. Stat. 28, No. 6, 1696--1701 (2000; Zbl 1105.62366)] and the observable Bessel process analyzed by \textit{P. Johnson} and \textit{G. Peskir} [Ann. Appl. Probab. 27, No. 2, 1003--1056 (2017; Zbl 1370.60135)] for the case of linear detection delay penalty.
Reviewer: Pavel Gapeev (London)On the \(\frac{S_n}{n}\) problemhttps://zbmath.org/1496.600362022-11-17T18:59:28.764376Z"Christensen, Sören"https://zbmath.org/authors/?q=ai:christensen.soren-gram|christensen.soren.2|christensen.soren-torholm|christensen.soren.1"Fischer, Simon"https://zbmath.org/authors/?q=ai:fischer.simon-raphaelSummary: The Chow-Robbins game is a classical, still partly unsolved, stopping problem introduced by \textit{Y. S. Chow} and \textit{H. Robbins} in [Ill. J. Math. 9, 444--454 (1965; Zbl 0173.46104)]. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads \(V(n,x)=\sup_{\tau}\mathbb{E} \left[\frac{x+S_\tau}{n+\tau}\right]\), where \(S\) is a random walk. We give a tight upper bound for \(V\) when \(S\) has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow-Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for \(n\leq 489 241\) .Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizonhttps://zbmath.org/1496.600372022-11-17T18:59:28.764376Z"De Angelis, Tiziano"https://zbmath.org/authors/?q=ai:de-angelis.tizianoSummary: We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them \textit{continuation bays} and \textit{stopping spikes}). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.On a switching control problem with càdlàg costshttps://zbmath.org/1496.600382022-11-17T18:59:28.764376Z"Hamadène, Said"https://zbmath.org/authors/?q=ai:hamadene.said"Jasso-Fuentes, Héctor"https://zbmath.org/authors/?q=ai:jasso-fuentes.hector"Osorio-Agudelo, Yamid A."https://zbmath.org/authors/?q=ai:osorio-agudelo.yamid-aSummary: This work addresses a switching control problem under which the cost associated with the changes of regimes is allowed to have discontinuities in time. Our main contribution is to show several characterizations of the optimal cost function as well as the existence of \(\varepsilon\)-optimal control policies. As a by-product, we also study the existence and uniqueness of solutions of a system of backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact of càdlàg type) and depend itself on the unknown solution. In the last part of the paper, we study the case when an underlying diffusion is part of the dynamic of the system. In this special case, the optimal payoff becomes a weak solution of the HJB system of PDEs with discontinuous obstacles which is of quasi-variational type. This paper is somehow a continuation of the papers [\textit{B. Djehiche} et al., SIAM J. Control Optim. 48, No. 4, 2751--2770 (2009; Zbl 1196.60069); \textit{S. Hamadène} and \textit{M. A. Morlais}, Appl. Math. Optim. 67, No. 2, 163--196 (2013; Zbl 1272.93130)] that consider continuous costs.Statistical causality and purely discontinuous local martingaleshttps://zbmath.org/1496.600392022-11-17T18:59:28.764376Z"Valjarević, Dragana"https://zbmath.org/authors/?q=ai:valjarevic.dragana"Petrović, Ljiljana"https://zbmath.org/authors/?q=ai:petrovic.ljiljanaSummary: The statistical concept of causality in continuous time between filtered probability spaces considered in this paper is based on Granger's definition of causality. The given concept of causality can be connected to the purely discontinuous property for martingale and filtration. If \(M\) is a purely discontinuous local martingale with respect to purely discontinuous filtration \((\mathcal{G}_t)\), we prove that \(M\) will remain purely discontinuous local martingale with respect to an extension \((\mathcal{F}_t)\) of the filtration \((\mathcal{G}_t)\) if and only if \((\mathcal{G}_t)\) is its own cause within \((\mathcal{F}_t)\). Moreover, we give conditions for a natural filtration of the purely discontinuous martingale \(X_t\) to be a purely discontinuous filtration, using the property of causality. We also, consider the connection between the concept of statistical causality and the weak predictable representation property for purely discontinuous local martingales.Some inferences on a 2- and 3-step random walkhttps://zbmath.org/1496.600402022-11-17T18:59:28.764376Z"Ahsanullah, M."https://zbmath.org/authors/?q=ai:ahsanullah.mohammad"Nevzorov, V. B."https://zbmath.org/authors/?q=ai:nevzorov.valerii-borisovichSummary: In 1905, Pearson proposed the following: ``A man starts from a point O and walks 1 step in a straight line, then he turns any angle whatever and walks 1 step in a straight line. He repeats this process \(n\) times. I require the probability that after \(n\) steps he is at a distance \(r\) and \(r + \mathrm{d}r\) from the starting point O.'' In this paper, we will present some basic properties and characterizations of the distribution of the distance for 2 and 3 steps.First-passage times for random walks in the triangular array settinghttps://zbmath.org/1496.600412022-11-17T18:59:28.764376Z"Denisov, Denis"https://zbmath.org/authors/?q=ai:denisov.denis-e"Sakhanenko, Alexander"https://zbmath.org/authors/?q=ai:sakhanenko.aleksandr-ivanovich"Wachtel, Vitali"https://zbmath.org/authors/?q=ai:wachtel.vitali-iSummary: In this paper we continue our study of exit times for random walks with independent but not necessarily identically distributed increments. Our paper [Ann. Probab. 46, No. 6, 3313--3350 (2018; Zbl 1434.60126)] was devoted to the case when the random walk is constructed by a fixed sequence of independent random variables which satisfies the classical Lindeberg condition. Now we consider a more general situation when we have a triangular array of independent random variables. Our main assumption is that the entries of every row are uniformly bounded by a deterministic sequence, which tends to zero as the number of the row increases.
For the entire collection see [Zbl 1478.60005].Angular asymptotics for random walkshttps://zbmath.org/1496.600422022-11-17T18:59:28.764376Z"López Hernández, Alejandro"https://zbmath.org/authors/?q=ai:hernandez.alejandro-lopez"Wade, Andrew R."https://zbmath.org/authors/?q=ai:wade.andrew-rSummary: We study the set of directions asymptotically explored by a spatially homogeneous random walk in \(d\)-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We also explore links to the asymptotics of one-dimensional projections, and to the growth of the convex hull of the random walk.
For the entire collection see [Zbl 1478.60005].Asymptotic formulas for the left truncated moments of sums with consistently varying distributed incrementshttps://zbmath.org/1496.600432022-11-17T18:59:28.764376Z"Sprindys, Jonas"https://zbmath.org/authors/?q=ai:sprindys.jonas"Šiaulys, Jonas"https://zbmath.org/authors/?q=ai:siaulys.jonasThe authors consider the sum \(S_n^\xi\) of possibly dependent and nonidentically distributed real-valued random variables with consistently varying distributions. This is a subclass of heavy-tailed distributions. By assuming that collection of the random variables follows the dependence structure that is similar to the asymptotic independence, more precisely, that they are pairwise quasi-asymptotically independent, the authors obtain asymptotic relations for the functionals of the form \(E((S_n^{\xi})^{\alpha}1_{S_n^\xi>x)})\) and \(E((S_n^{\xi}-x)^+)^{\alpha}\), where \(\alpha\) is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.
Reviewer: Yuliya S. Mishura (Kyïv)On Doney's striking factorization of the arc-sine lawhttps://zbmath.org/1496.600442022-11-17T18:59:28.764376Z"Alili, Larbi"https://zbmath.org/authors/?q=ai:alili.larbi"Bartholmé, Carine"https://zbmath.org/authors/?q=ai:bartholme.carine"Chaumont, Loïc"https://zbmath.org/authors/?q=ai:chaumont.loic"Patie, Pierre"https://zbmath.org/authors/?q=ai:patie.pierre"Savov, Mladen"https://zbmath.org/authors/?q=ai:savov.mladen-svetoslavov"Vakeroudis, Stavros"https://zbmath.org/authors/?q=ai:vakeroudis.stavros|vakeroudis.stavros-mSummary: In [Bull. Lond. Math. Soc. 19, 177--182 (1987; Zbl 0591.60066)], \textit{R. A. Doney} identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.
For the entire collection see [Zbl 1478.60005].Independent increment processes: a multilinearity preserving propertyhttps://zbmath.org/1496.600452022-11-17T18:59:28.764376Z"Benth, Fred Espen"https://zbmath.org/authors/?q=ai:benth.fred-espen"Detering, Nils"https://zbmath.org/authors/?q=ai:detering.nils"Krühner, Paul"https://zbmath.org/authors/?q=ai:kruhner.paulSummary: We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space \(B\). It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of \(B\) being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.A transformation for spectrally negative Lévy processes and applicationshttps://zbmath.org/1496.600462022-11-17T18:59:28.764376Z"Chazal, Marie"https://zbmath.org/authors/?q=ai:chazal.marie"Kyprianou, Andreas E."https://zbmath.org/authors/?q=ai:kyprianou.andreas-e"Patie, Pierre"https://zbmath.org/authors/?q=ai:patie.pierreSummary: The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts,
[the last author, Bull. Sci. Math. 133, No. 4, 355--382 (2009; Zbl 1171.60009); Bernoulli 17, No. 2, 814--826 (2011; Zbl 1253.60020); the second and third author, Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 3, 917--928 (2011; Zbl 1231.60031); \textit{A. V. Gnedin}, ``Regeneration in random combinatorial structures'', Preprint, \url{arXiv:0901.4444}; the last author and \textit{M. Savov}, Electron. J. Probab. 17, Paper No. 38, 22 p. (2012; Zbl 1253.60063)], as well as in older work of
\textit{K. Urbanik} [Probab. Math. Stat. 15, 493--513 (1995; Zbl 0852.60015)].
We show how some specific instances of this mapping prove to be useful for a variety of applications.
For the entire collection see [Zbl 1478.60005].A dual Yamada-Watanabe theorem for Lévy driven stochastic differential equationshttps://zbmath.org/1496.600472022-11-17T18:59:28.764376Z"Criens, David"https://zbmath.org/authors/?q=ai:criens.davidIn this article the author establishes the dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations (SDEs) driven by Lévy noise. More precisely, let \(L\) be a quasi-left continuous semimartingale with independent increments, let \(\mu\) and \(\sigma\) be real-valued predictable processes on the path space of real-valued càdlàg functions, and consider the real-valued SDE \[ d X_t = \mu_t(X) dt + \sigma_t(X) dL_t, \quad X_0 = x_0 \in \mathbb{R}. \] Then the main result of the paper (see Theorem 1.1) states that the following statements are equivalent:
\begin{itemize}
\item[(i)] Strong uniqueness and weak existence holds.
\item[(ii)] Weak uniqueness and strong existence holds.
\item[(iii)] Weak joint uniqueness and strong existence holds.
\end{itemize}
This result closes a gap in the literature. As pointed out by the author, so far the only article dealing with the dual Yamada-Watanabe theorem with discontinuous noise was [\textit{H. Zhao} et al., ``Equivalence of uniqueness in law and joint uniqueness in law for SDEs driven by Poisson processes'', Appl. Math. 7, No. 8, 784--792 (2016; \url{doi:10.4236/am.2016.78070})], and there was a gap the proof.
Reviewer: Stefan Tappe (Freiburg)Two continua of embedded regenerative setshttps://zbmath.org/1496.600482022-11-17T18:59:28.764376Z"Evans, Steven N."https://zbmath.org/authors/?q=ai:evans.steven-neil"Ouaki, Mehdi"https://zbmath.org/authors/?q=ai:ouaki.mehdiSummary: Given a two-sided real-valued Lévy process \((X_t)_{t \in \mathbb{R}}\), define processes \((L_t)_{t \in \mathbb{R}}\) and \((M_t)_{t \in \mathbb{R}}\) by \(L_t := \sup \{h \in \mathbb{R} : h - \alpha (t-s) \le X_s \text{ for all } s \le t\} = \inf \{X_s + \alpha (t-s) : s \le t\}\), \(t \in \mathbb{R}\), and \(M_t := \sup \{ h \in \mathbb{R} : h - \alpha |t-s| \leq X_s \text{ for all } s \in \mathbb{R} \} = \inf \{X_s + \alpha |t-s| : s \in \mathbb{R}\}\), \(t \in \mathbb{R}\). The corresponding contact sets are the random sets \(\mathcal{H}_\alpha := \{ t \in \mathbb{R} : X_t\wedge X_{t-} = L_t\}\) and \(\mathcal{Z}_\alpha := \{ t \in \mathbb{R} : X_t\wedge X_{t-} = M_t\}\). For a fixed \(\alpha >\mathbb{E}[X_1]\) (resp. \(\alpha >|\mathbb{E}[X_1]|\)) the set \(\mathcal{H}_\alpha\) (resp. \(\mathcal{Z}_\alpha\)) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections \((\mathcal{H}_{\alpha })_{\alpha > \mathbb{E}[X_1]}\) and \((\mathcal{Z}_{\alpha })_{\alpha > |\mathbb{E}[X_1]|}\) are increasing in \(\alpha\) and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that \((\sup \{t < 0 : t \in \mathcal{H}_\alpha \})_{\alpha > \mathbb{E}[X_1]}\) is a càdlàg, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for \((\sup \{t < 0 : t \in \mathcal{Z}_\alpha \})_{\alpha > |\beta |}\) when \((X_t)_{t \in \mathbb{R}}\) is a (two-sided) Brownian motion with drift \(\beta\).
For the entire collection see [Zbl 1478.60005].Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processeshttps://zbmath.org/1496.600492022-11-17T18:59:28.764376Z"Kwaśnicki, Mateusz"https://zbmath.org/authors/?q=ai:kwasnicki.mateusz"Kyprianou, Andreas E."https://zbmath.org/authors/?q=ai:kyprianou.andreas-e"Palau, Sandra"https://zbmath.org/authors/?q=ai:palau.sandra"Saizmaa, Tsogzolmaa"https://zbmath.org/authors/?q=ai:saizmaa.tsogzolmaaSummary: Suppose that \(\mathsf{S}\) is a closed set of the unit sphere \(\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}\) in dimension \(d\geq 2\), which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension \(d\geq 2\) conditioned to approach \(\mathsf{S}\) continuously, allowing for the interior and exterior of \(\mathbb{S}^{d-1}\) to be visited infinitely often. Additionally, we show that this process is in duality with the unconditioned stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that \(\mathsf{S}\) is replaced by \(\mathsf{D}\), a closed bounded subset of the hyperplane \(\{x\in \mathbb{R}^d : (x, v) = 0\}\) with positive surface measure, where \(v\) is the unit orthogonal vector and where \((\cdot, \cdot)\) is the usual Euclidean inner product. Our results complement similar results of the last three authors [Stochastic Processes Appl. 137, 272--293 (2021; Zbl 1469.60149)] in which the stable process was further constrained to attract to and repel from \(\mathsf{S}\) from either the exterior or the interior of the unit sphere.
For the entire collection see [Zbl 1478.60005].The Doob-McKean identity for stable Lévy processeshttps://zbmath.org/1496.600502022-11-17T18:59:28.764376Z"Kyprianou, Andreas E."https://zbmath.org/authors/?q=ai:kyprianou.andreas-e"O'Connell, Neil"https://zbmath.org/authors/?q=ai:oconnell.neilSummary: We re-examine the celebrated Doob-McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic \(\alpha\)-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic \(\alpha\)-stable process, providing one interprets the components of the original identity in the right way.
For the entire collection see [Zbl 1478.60005].Asymptotic behaviour on the linear self-interacting diffusion driven by \(\alpha\)-stable motionhttps://zbmath.org/1496.600512022-11-17T18:59:28.764376Z"Sun, Xichao"https://zbmath.org/authors/?q=ai:sun.xichao"Yan, Litan"https://zbmath.org/authors/?q=ai:yan.litanSummary: In this paper, as an attempt we consider the linear self-interacting diffusion driven by an \(\alpha\)-stable motion, which is the solution to the equation
\[
X^\alpha_t = M^\alpha_t -\theta \int^t_0\int^s_0(X^\alpha_s -X^\alpha_r) \mathrm{d}r \mathrm{d}s +
\nu t,
\]
where \(\theta \neq 0\), \(\nu \in \mathbb{R}\) and \(M^\alpha\) is an \(\alpha\)-stable motion on \(\mathbb{R} (0 <\alpha \leq 2)\). The process is an analogue of the self-attracting diffusion (see [\textit{R. T. Durrett} and \textit{L. C. G. Rogers}, Probab. Theory Relat. Fields 92, No. 3, 337--349 (1992; Zbl 0767.60080)] and [\textit{M. Cranston} and \textit{Y. Le Jan}, Math. Ann. 303, No. 1, 87--93 (1995; Zbl 0838.60052)]). The main object of this paper is to prove some limit theorems associated with the solution process \(X^\alpha\) for \(\frac{1}{2} < \alpha \leq 2\). When \(\theta > 0\) we show that \(\psi_\alpha (t) (X^\alpha_t - X^\alpha_\infty)\) converges to an \(\alpha\)-stable random variable in distribution, as \(t\) tends to infinity, where \(\psi_\alpha (t) = t^{1/\alpha}\) for \(1 \leq \alpha \leq 2\) and \(\psi_\alpha (t) = t^{2-\frac{1}{\alpha}}\) for \(\frac{1}{2} < \alpha < 1\). When \(\theta < 0\), for all \(\frac{1}{2} < \alpha \leq 2\) we show that, as \(t \to \infty\), \(J^\alpha_t (\theta , \nu , 0) := te^{\frac{1}{2} \theta t^2} X^\alpha_t\) converges to \(\xi^\alpha_\infty - \frac{\nu}{\theta}\) and
\[
\begin{aligned}
J^\alpha_t (\theta , \nu , n) : = &-\theta t^2 \left( J^\alpha_t (\theta , \nu , n -1) -(2n -3)!! \left( \xi^\alpha_\infty - \frac{\nu}{\theta} \right) \right)\\
&\to (2n -1)!! \left( \xi^\alpha_\infty - \frac{\nu}{\theta} \right)
\end{aligned}
\]
a.s. for all \(n \geq 1\), where \((-1)!! = 1\) and \(\xi^\alpha_\infty = \int^\infty_0 se^{\frac{1}{2} \theta s^2} \mathrm{d}M^\alpha_s\).Nonlinear filtering of stochastic differential equations with correlated Lévy noiseshttps://zbmath.org/1496.600522022-11-17T18:59:28.764376Z"Qiao, Huijie"https://zbmath.org/authors/?q=ai:qiao.huijieSummary: The work concerns nonlinear filtering problems of stochastic differential equations with correlated Lévy noises. First, we establish the Kushner-Stratonovich and Zakai equations through martingale representation theorems and the Kallianpur-Striebel formula. Second, we show the pathwise uniqueness and uniqueness in joint law of weak solutions for the Zakai equation. Finally, we investigate the uniqueness in joint law of weak solutions to the Kushner-Stratonovich equation.Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependencehttps://zbmath.org/1496.600532022-11-17T18:59:28.764376Z"Makogin, Vitalii"https://zbmath.org/authors/?q=ai:makogin.vitalii"Spodarev, Evgeny"https://zbmath.org/authors/?q=ai:spodarev.evgueniSummary: This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.Large degrees in scale-free inhomogeneous random graphshttps://zbmath.org/1496.600542022-11-17T18:59:28.764376Z"Bhattacharjee, Chinmoy"https://zbmath.org/authors/?q=ai:bhattacharjee.chinmoy"Schulte, Matthias"https://zbmath.org/authors/?q=ai:schulte.matthiasThe authors study the large degrees for a general class of scale-free inhomogeneous random graph models, which includes Norros-Reittu model and the scale-free continuum percolation model. For increasing observation windows, the authors prove that the maximum degree in such graphs, after rescaling, converges to a Frechet-distrubuted random variable, by showing the convergence of the rescaled degree sequences of the random graphs to a Poisson process. The authors prove the consistency of the Hill estimator for the inverse of the tail exponent of the power-law tail of the typical degree distribution.
Reviewer: Ravi Sreenivasan (Mysore)Extremes and regular variationhttps://zbmath.org/1496.600552022-11-17T18:59:28.764376Z"Bingham, Nick H."https://zbmath.org/authors/?q=ai:bingham.nicholas-h"Ostaszewski, Adam J."https://zbmath.org/authors/?q=ai:ostaszewski.adam-jSummary: We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the point of view of our recent work on general regular variation.
For the entire collection see [Zbl 1478.60005].No-tie conditions for large values of extremal processeshttps://zbmath.org/1496.600562022-11-17T18:59:28.764376Z"Ipsen, Yuguang"https://zbmath.org/authors/?q=ai:ipsen.yuguang-f"Maller, Ross"https://zbmath.org/authors/?q=ai:maller.ross-arthurSummary: We give necessary and sufficient conditions for there to be no ties, asymptotically, among large values of a space-time Poisson point process evolving homogeneously in time. The convergence is at small times, in probability or almost sure.
For the entire collection see [Zbl 1478.60005].On the lack of semimartingale propertyhttps://zbmath.org/1496.600572022-11-17T18:59:28.764376Z"Prokaj, Vilmos"https://zbmath.org/authors/?q=ai:prokaj.vilmos"Bondici, László"https://zbmath.org/authors/?q=ai:bondici.laszloThe authors extend the characterization of semimartingale functions due to \textit{E. Cinlar} et al. [Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 161--219 (1980; Zbl 0443.60074)] to the non-Markovian setting. They show that if a function of a semimartingale remains a semimartingale, then under certain conditions the function must have intervals where it is a difference of two convex functions. Under suitable conditions this property also holds for random functions. As an application, they prove that the median process defined by \textit{V. Prokaj} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 2, 498--514 (2011; Zbl 1216.60048)] is not a semimartingale. This is defined as follows. Let \((D_t(x))_{t\geq 0}\) be the solution of \(d D_t(x) = \sigma(D_t(x))dB_t\), \(D_0(x)=x\in[0,1]\), where \(\sigma(x)=x\wedge(1-x)\). The median process is \((D^{-1}_t(1/2))_{t\geq 0}\). More generally, they show that \((D^{-1}_t (\alpha))_{t\geq 0}\) is not a semimartingale for \(\alpha\in(0, 1)\).
Reviewer: Franco Fagnola (Milano)Atypical exit events near a repelling equilibriumhttps://zbmath.org/1496.600582022-11-17T18:59:28.764376Z"Bakhtin, Yuri"https://zbmath.org/authors/?q=ai:bakhtin.yuri-yu"Chen, Hong-Bin"https://zbmath.org/authors/?q=ai:chen.hongbinConsider a stochastic diffusion process \(X\) with a small diffusion coefficient of order \(\varepsilon\). Assume that the associated dynamical system obtained in the limit \(\varepsilon=0\) has an unstable equilibrium point and that the stochastic system starts from this point. We know from the classical Freidlin-Wentzell theory that the probability of quitting a neighbourhood of this equilibrium before some fixed time is exponentially small, with a rate of order \(1/\varepsilon^2\). However, in longer times, the process will eventually quit this neighbourhood at a time \(\tau\). It was known that the escape time is of order \(\log(1/\varepsilon)\), and that the exit location \(X_\tau\) is obtained by following the most unstable direction(s). The aim of the paper under review is to consider more unlikely exit locations occurring in the case of an abnormally long stay in the neighbourhood.
The main result states that under convenient assumptions, \(\mathbb{P}(X_\tau\in A)\sim\varepsilon^{\rho(A)}\mu(A)\) for \(\rho(A)\) taking values in a discrete set depending on the geometry of the dynamical system. Moreover, the conditional distribution of \(X_\tau\) given \(\{X_\tau\in A\}\) is studied.
For the proof one notices that it is sufficient to consider arbitrarily small neighbourhoods, since after quitting such a neighbourhood the process is almost deterministic (due to the unstability of the equilibrium). On the other hand, on a small neighbourhood the process can be approximated by a Gaussian process. The approximation error was already studied by the same authors by applying Malliavin's calculus, so they have to study the behaviour of the Gaussian process.
Reviewer: Jean Picard (Aubière)On comparison theorem for optional SDEs via local times and applicationshttps://zbmath.org/1496.600592022-11-17T18:59:28.764376Z"Abdelghani, Mohamed"https://zbmath.org/authors/?q=ai:abdelghani.mohamed-n"Melnikov, Alexander"https://zbmath.org/authors/?q=ai:melnikov.alexander-v"Pak, Andrey"https://zbmath.org/authors/?q=ai:pak.andreySummary: In this paper, we study SDEs with respect to optional semimartingales or optional SDEs. Our leading idea is to explore the concept and technique of local time of optional processes to extend several results on comparison and pathwise uniqueness of solutions of such stochastic equations. We also obtain a comparison result for optional stochastic equations with different jump-diffusions. Moreover, we apply our comparison theorem to calculate option price bounds in mathematical finance. Our findings are supported by numerical examples.A transformation method to study the solvability of fully coupled FBSDEshttps://zbmath.org/1496.600602022-11-17T18:59:28.764376Z"Ankirchner, Stefan"https://zbmath.org/authors/?q=ai:ankirchner.stefan"Fromm, Alexander"https://zbmath.org/authors/?q=ai:fromm.alexander"Wendt, Julian"https://zbmath.org/authors/?q=ai:wendt.julianSummary: We consider fully coupled forward-backward stochastic differential equations (FBSDEs), where all function parameters are Lipschitz continuous, the terminal condition is monotone and the diffusion coefficient of the forward part depends monotonically on \(z\), the control process component of the backward part. We show that there exists a class of linear transformations turning the FBSDE into an auxiliary FBSDE for which the Lipschitz constant of the forward diffusion coefficient w.r.t. \(z\) is smaller than the inverse of the Lipschitz constant of the terminal condition w.r.t. the forward component \(x\). The latter condition allows to verify existence of a global solution by analysing the spatial derivative of the decoupling field. This is useful since by applying the inverse linear transformation to a solution of the auxiliary FBSDE we obtain a solution to the original one. We illustrate with several examples how linear transformations, combined with an analysis of the decoupling field's gradient, can be used for proving global solvability of FBSDEs.Convergence rate of the EM algorithm for SDEs with low regular driftshttps://zbmath.org/1496.600612022-11-17T18:59:28.764376Z"Bao, Jianhai"https://zbmath.org/authors/?q=ai:bao.jianhai"Huang, Xing"https://zbmath.org/authors/?q=ai:huang.xing"Zhang, Shao-Qin"https://zbmath.org/authors/?q=ai:zhang.shaoqinSummary: In this paper we employ a Gaussian-type heat kernel estimate to establish Krylov's estimate and Khasminskii's estimate for the Euler-Maruyama (EM) algorithm. For applications, by taking Zvonkin's transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.Regularization effects of a noise propagating through a chain of differential equations: an almost sharp resulthttps://zbmath.org/1496.600622022-11-17T18:59:28.764376Z"De Raynal, Paul-Éric Chaudru"https://zbmath.org/authors/?q=ai:chaudru-de-raynal.paul-eric"Menozzi, Stéphane"https://zbmath.org/authors/?q=ai:menozzi.stephaneSummary: We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like Hörmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counter-examples, almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.Asymptotic moment estimation for stochastic Lotka-Volterra model driven by \(G\)-Brownian motionhttps://zbmath.org/1496.600632022-11-17T18:59:28.764376Z"He, Ping"https://zbmath.org/authors/?q=ai:he.ping"Ren, Yong"https://zbmath.org/authors/?q=ai:ren.yong"Zhang, Defei"https://zbmath.org/authors/?q=ai:zhang.defeiSummary: A stochastic Lotka-Volterra model disturbed by \(G\)-Brownian motion (G-LVM for short) in the framework of non-linear expectation is proposed in this paper. This model takes into account the uncertainty of variance of the noise. We prove the G-LVM exists a unique solution and the solution does not tend to infinity when the time is finite under some constraints, and obtain many asymptotic moment estimations which depend on the variance of \(G\)-Brownian motion by capacity theory, exponential martingale inequality and analytical skills.Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulseshttps://zbmath.org/1496.600642022-11-17T18:59:28.764376Z"Liu, Jiankang"https://zbmath.org/authors/?q=ai:liu.jiankang"Xu, Wei"https://zbmath.org/authors/?q=ai:xu.wei.1"Guo, Qin"https://zbmath.org/authors/?q=ai:guo.qin"Wang, Jinbin"https://zbmath.org/authors/?q=ai:wang.jinbinSummary: This paper aims at addressing the issue of a periodic averaging method for neutral stochastic functional differential equations with delayed impulses. Two periodic averaging theorems are presented and the approximate equivalence between the solutions to the original systems and those to the reduced averaged systems without impulses is proved. Further, we show a brief framework of extending our main results to Lévy case. At last, an example is given to demonstrate the procedure and validity of the proposed periodic averaging method.Almost periodic and periodic solutions of differential equations driven by the fractional Brownian motion with statistical applicationhttps://zbmath.org/1496.600652022-11-17T18:59:28.764376Z"Marie, Nicolas"https://zbmath.org/authors/?q=ai:marie.nicolas"Raynaud de Fitte, Paul"https://zbmath.org/authors/?q=ai:raynaud-de-fitte.paulSummary: We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.The Magnus expansion for stochastic differential equationshttps://zbmath.org/1496.600662022-11-17T18:59:28.764376Z"Wang, Zhenyu"https://zbmath.org/authors/?q=ai:wang.zhenyu.1"Ma, Qiang"https://zbmath.org/authors/?q=ai:ma.qiang"Yao, Zhen"https://zbmath.org/authors/?q=ai:yao.zhen"Ding, Xiaohua"https://zbmath.org/authors/?q=ai:ding.xiaohuaSummary: In this paper, all the terms in the stochastic Magnus expansion are presented by rooted trees. First, stochastic Magnus methods for linear stochastic differential equations are constructed by truncating the stochastic Magnus expansion. Then, explicit stochastic Magnus methods are constructed by Picard's iteration for nonlinear stochastic differential equations on matrix Lie group. Furthermore, general nonlinear stochastic differential equations are transformed into linear operator stochastic differential equations by using the Lie derivative. Finally, numerical methods for general nonlinear stochastic differential equations are constructed by using the theory of the stochastic Magnus expansion for the linear case. In particular, for the commutative case, it is shown that the stochastic Magnus expansion provides a novel way to construct computationally inexpensive and arbitrarily high-order numerical methods while avoiding the simulation of multiple stochastic integrals. Moreover, the proposed methods are shown to preserve the intrinsic properties of the original system well and the numerical experiments agree with the theoretical results.Stochastic averaging principle for two-time-scale jump-diffusion SDEs under the non-Lipschitz coefficientshttps://zbmath.org/1496.600672022-11-17T18:59:28.764376Z"Xu, Jie"https://zbmath.org/authors/?q=ai:xu.jie"Liu, Jicheng"https://zbmath.org/authors/?q=ai:liu.jichengSummary: In this paper, we shall prove a stochastic averaging principle for two-time-scale jump-diffusion SDEs under the non-Lipschitz coefficients.Nonlinear parabolic stochastic evolution equations in critical spaces. I: Stochastic maximal regularity and local existencehttps://zbmath.org/1496.600682022-11-17T18:59:28.764376Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cAuthors' abstract: In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an \(L^p\)-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers' equation, the Allen-Cahn equation, the Cahn-Hilliard equation, reaction-diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to \textit{J. Prüss} et al. [J. Differ. Equations 264, No. 3, 2028--2074 (2018; Zbl 1377.35176)]. Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.
Reviewer: Piotr Biler (Wrocław)Spatial integral of the solution to hyperbolic Anderson model with time-independent noisehttps://zbmath.org/1496.600692022-11-17T18:59:28.764376Z"Balan, Raluca M."https://zbmath.org/authors/?q=ai:balan.raluca-m"Yuan, Wangjun"https://zbmath.org/authors/?q=ai:yuan.wangjunSummary: In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension \(d \leq 2\), as the domain of the integral gets large (for fixed time \(t)\). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.On backward SPDEs without proper Cauchy conditionhttps://zbmath.org/1496.600702022-11-17T18:59:28.764376Z"Dokuchaev, Nikolai"https://zbmath.org/authors/?q=ai:dokuchaev.nikolai-gSummary: We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected.Limiting dynamics for stochastic nonclassical diffusion equationshttps://zbmath.org/1496.600712022-11-17T18:59:28.764376Z"Gao, Peng"https://zbmath.org/authors/?q=ai:gao.peng.1|gao.pengSummary: In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.The averaging method for doubly perturbed distribution dependent SDEshttps://zbmath.org/1496.600722022-11-17T18:59:28.764376Z"Ma, Xiaocui"https://zbmath.org/authors/?q=ai:ma.xiaocui"Yue, Haitao"https://zbmath.org/authors/?q=ai:yue.haitao"Xi, Fubao"https://zbmath.org/authors/?q=ai:xi.fubaoThis work studies the averaging method for the so-called doubly perturbed distribution dependent SDEs (DPDSDE). The averaging principle for this DPDSDEs is considered, and an approximation result is established.
The well-posedness of the DPDSDEs is also proved using the fixed point theorem. It was stated that the solutions of the original equations converge to those of the averaged equations in the sense of mean square and probability. An example is provided to show the applications of the results under consideration.
Reviewer: Anatoliy Swishchuk (Calgary)Lie symmetry analysis on pricing weather derivatives by partial differential equationshttps://zbmath.org/1496.600732022-11-17T18:59:28.764376Z"Nhangumbe, Clarinda"https://zbmath.org/authors/?q=ai:nhangumbe.clarinda"Fredericks, Ebrahim"https://zbmath.org/authors/?q=ai:fredericks.ebrahim"Canhanga, Betuel"https://zbmath.org/authors/?q=ai:canhanga.betuelRecently, stochastic differential equations have been proposed to model some weather derivatives. In this paper, the authors shed light on the irregularity of the rainfall intensity and the duration of the dry spells represented by the mean reverting Ornstein-Uhlenbeck process \((X_t)_t\) satisfying the stochastic differential equation:
\[
dX_t=[k (\theta(t)-X_t)+\theta'(t)]dt+\sigma_tdW_t,\tag{1}
\]
where \(k\) is the rate of the mean reversion, \(\sigma_t\) is the volatility of the rainfall, \((W_t)_t\) is a Brownian motion, \(\theta(.)\) is the long term mean of the process \(X_t\) given by:
\[
\theta(t)= m+\alpha \sin\big( \frac{\pi(t-v)}{6}\big)\tag{2}
\]
with \(v\) being the shift of the \(X-\)axis, \(m\) being the mean of the sine curve and \(\alpha\) determining the oscillation. Using Feynman-Kac Theorem, the underlying value of the weather option \(V(x, y, t)\) should satisfy the following partial differential equation (3):
\[
\frac{\partial V}{\partial t}= r V- f(x)\frac{\partial V}{\partial y}-[k (\theta(t)-x)+\theta'(t)-\lambda\sigma_t]\frac{\partial V}{\partial x}-\frac12 \sigma_t^2\frac{\partial^2 V}{\partial^2x}.\tag{3}
\]
The equation is associated with the terminal condition: \( V(x, y, T)= \mathrm{tick}\times (S-Y(T))^+ \), where \( f(x)\) is the pay off function, \(\lambda\) is the market price of the risk, \(y(.)\) is the European input value of the Rain Defice Day (RDD), \(S\) is the strike level and ``tick'' is used to convert into monetary terms.
Although the application of the Lie group classification show that the Eq. (3) cannot be reduced to a heat equation for any values of the parameters, the authors find an only two symmetries subalgebra that is used to reduce Eq. (3) by an independent variable. They determine, through Yu, Li and Chen algorithm, the one-dimensional optimal system of the algebra. The optimal system allows to gather invariant solutions into equivalent classes.
For the entire collection see [Zbl 1467.16001].
Reviewer: Latifa Debbi (M'Sila)Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroupshttps://zbmath.org/1496.600742022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Abdelmonem, Ahmed"https://zbmath.org/authors/?q=ai:abdelmonem.ahmed"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.RETRACTED ARTICLE: Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motionshttps://zbmath.org/1496.600752022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo|sun.jianguo.1"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: By using new Schrödinger type inequalities appearing in [\textit{Z. Jiang} and \textit{F. M. Usó}, J. Inequal. Appl. 2016, Paper No. 233, 10 p. (2016; Zbl 1350.35070)], we study the existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions.
Editorial remark. This article has been retracted. According to the retraction notice [\textit{J. Sun} et al., J. Inequal. Appl. 2021, Paper No. 109, 1 p. (2021; Zbl 1496.60076)], ``the Editors-in-Chief have retracted this article because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Retraction note: ``Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions''https://zbmath.org/1496.600762022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo.1|sun.jianguo"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: The Editors-in-Chief have retracted this article [\textit{J. Sun}, ibid. 2018, Paper No. 100, 15 p. (2018; Zbl 1496.60075)] because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Slowly varying asymptotics for signed stochastic difference equationshttps://zbmath.org/1496.600772022-11-17T18:59:28.764376Z"Korshunov, Dmitry"https://zbmath.org/authors/?q=ai:korshunov.dmitrySummary: For a stochastic difference equation \(D_n = A_n D_{n -1} + B_n\) which stabilises upon time we study tail distribution asymptotics for \(D_n\) under the assumption that the distribution of \(\log (1+|A_1|+|B_1|)\) is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the present paper is three-fold. Firstly, we identify the asymptotic behaviour not only of the stationary tail distribution but also of \(D_n\). Secondly, we solve the problem in the general setting when A takes both positive and negative values. Thirdly, we get rid of auxiliary conditions like finiteness of higher moments introduced in the literature before.
For the entire collection see [Zbl 1478.60005].Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noisehttps://zbmath.org/1496.600782022-11-17T18:59:28.764376Z"Chang, Yanqiang"https://zbmath.org/authors/?q=ai:chang.yanqiang"Chen, Huabin"https://zbmath.org/authors/?q=ai:chen.huabinSummary: In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by Lévy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in \(p\)th\((p\geq 2)\) for stochastic delay differential equations with Markovian switching driven by Lévy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.On the validity of the Girsanov transformation method for sensitivity analysis of stochastic chemical reaction networkshttps://zbmath.org/1496.600792022-11-17T18:59:28.764376Z"Wang, Ting"https://zbmath.org/authors/?q=ai:wang.ting"Rathinam, Muruhan"https://zbmath.org/authors/?q=ai:rathinam.muruhanSummary: We investigate the validity of the Girsanov Transformation (GT) method for parametric sensitivity analysis of stochastic models of chemical reaction networks. The validity depends on the likelihood ratio process being a martingale and the commutation of a certain derivative with expectation. We derive some exponential integrability conditions which imply both these requirements. We provide further conditions in terms of a reaction network that imply these exponential integrability conditions.The unified colored noise approximation of multidimensional stochastic dynamic systemhttps://zbmath.org/1496.600802022-11-17T18:59:28.764376Z"Duan, Wei-Long"https://zbmath.org/authors/?q=ai:duan.weilong"Fang, Hui"https://zbmath.org/authors/?q=ai:fang.huiSummary: The unified colored noise approximation of multidimensional stochastic dynamic system driven by correlated Gaussian colored noises is developed. For general multidimensional stochastic dynamic system driven by Gaussian colored noises, which maybe correlate with each other and have small or large correlation times, it is transformed into multidimensional stochastic dynamic system driven by white noises. By merging method of white noises, the multiple diffusion terms of each equation of system is merged into one diffusion term, then the unified colored noise approximation for multidimensional stochastic dynamic system is established. In order to facilitate application, the specific derivation formulas of the unified colored noise approximation for common one-dimensional, two-dimensional, and three-dimensional stochastic dynamic systems are also given respectively, among, one also proves that it includes the unified colored noise approximation for one-dimensional stochastic dynamic system previously created.Quantization methods for stochastic differential equationshttps://zbmath.org/1496.600812022-11-17T18:59:28.764376Z"Kienitz, J."https://zbmath.org/authors/?q=ai:kienitz.jorg"McWalter, T. A."https://zbmath.org/authors/?q=ai:mcwalter.thomas-andrew"Rudd, R."https://zbmath.org/authors/?q=ai:rudd.robert-e"Platen, E."https://zbmath.org/authors/?q=ai:platen.eckhardSummary: In this paper we provide an introduction to quantization with applications in quantitative finance. We start with a review of vector quantization (VQ), a method originally devised for lossy signal compression. The basics of VQ are presented and applied to probability distributions. n order to solve stochastic differential equations (SDEs), a recursive algorithm based on the Euler-Maruyama approximation was devised by \textit{G. Pagès} and \textit{A. Sagna} [Appl. Math. Finance 22, No. 5--6, 463--498 (2015; Zbl 1396.91805)]. The approach, known as recursive marginal quantization is described. We show how RMQ can be adapted to ensure non-negativity of solutions. We further provide the basic details on how higher-order Itō-Taylor schemes may be implemented within the RMQ framework. Then, having treated the case of one-dimensional SDEs, we turn our attention to the coupled SDEs used in stochastic volatility models. We present Joint RMQ, an instance of a product quantizer. Finally we demonstrate the performance of the methods with some financial pricing problems. In particular we show the performance improvements of the higher order RMQ methods and provide numerical examples of contingent claim pricing for the quadratic volatility, Stein-Stein and Heston models. This paper is styled as a ``guided tour'' throughout -- introducing the reader to the basic concepts and then referring to the original sources that provide full detail.
For the entire collection see [Zbl 1483.65008].Kernel-based collocation methods for Heath-Jarrow-Morton models with Musiela parametrizationhttps://zbmath.org/1496.600822022-11-17T18:59:28.764376Z"Kinoshita, Yuki"https://zbmath.org/authors/?q=ai:kinoshita.yuki"Nakano, Yumiharu"https://zbmath.org/authors/?q=ai:nakano.yumiharuSummary: We propose kernel-based collocation methods for numerical solutions to Heath-Jarrow-Morton models with Musiela parametrization. The methods can be seen as the Euler-Maruyama approximation of some finite-dimensional stochastic differential equations and allow us to compute the derivative prices by the usual Monte Carlo methods. We derive a bound on the rate of convergence under some decay conditions on the interpolation functions and some regularity conditions on the volatility functionals.Pathwise stability and positivity of semi-discrete approximations of the solution of nonlinear stochastic differential equationshttps://zbmath.org/1496.600832022-11-17T18:59:28.764376Z"Stamatiou, Ioannis S."https://zbmath.org/authors/?q=ai:stamatiou.ioannis-sSummary: We use the main idea of the semi-discrete method, originally proposed in [\textit{N. Halidias}, Int. J. Comput. Math. 89, No. 6, 780--794 (2012; Zbl 1255.65020)], to reproduce qualitative properties of a class of nonlinear stochastic differential equations with non-negative, non-globally Lipschitz coefficients and a unique equilibrium solution. The proposed fixed-time step method preserves the positivity of the solution and reproduces the almost sure asymptotic stability behavior of the equilibrium with no time-step restrictions. In particular, we are interested in the following class of scalar stochastic differential equations,
\[
x_t =x_0 + \int_0^t x_sa(x_s)ds + \int_0^t x_sb(x_s)dW_s,
\]
where \(a (\cdot)\), \(b (\cdot)\) are non-negative functions with \(b(u) \neq 0\) for \(u \neq 0\), \(x_0 \geq 0\) and \(\{W_t\}_{t \geq 0}\) is a one-dimensional Wiener process adapted to the filtration \(\{{\mathcal F}_t\}_{t\geq 0}\).
For the entire collection see [Zbl 1483.00042].Algorithmic schemes for classes of quadratic stochastic operators and partial differential evolution equations corresponding to themhttps://zbmath.org/1496.600842022-11-17T18:59:28.764376Z"Normatov, I. Kh."https://zbmath.org/authors/?q=ai:normatov.i-kh(no abstract)Stationary distributions and ergodicity of reflection-type Markov chainshttps://zbmath.org/1496.600852022-11-17T18:59:28.764376Z"Liu, Yujie"https://zbmath.org/authors/?q=ai:liu.yujie"Niu, Minwen"https://zbmath.org/authors/?q=ai:niu.minwen"Yao, Dacheng"https://zbmath.org/authors/?q=ai:yao.dacheng"Zhang, Hanqin"https://zbmath.org/authors/?q=ai:zhang.hanqinThe paper considers a so-called reflection-type Markov chain (RTMC). This RTMC is usually used to characterize some process features in stochastic control and operations management.
Regarding that this RTMC is the Feller chain, the existence of its stationary distributions is proved. Also, ergodicity of this RTMC is obtained by constructing the small set. Moreover, for the non-ergodic case, the system dynamics are completely described.
Reviewer: Anatoliy Swishchuk (Calgary)Counting the zeros of an elephant random walkhttps://zbmath.org/1496.600862022-11-17T18:59:28.764376Z"Bertoin, Jean"https://zbmath.org/authors/?q=ai:bertoin.jeanThe elephant random walk (ERW) can be viewed as a member of the family of reinforced processes (see [\textit{G. M. Schütz} and \textit{S. Trimper}, ``Elephants can always remember: exact long-range memory effects in a non-Markovian random walk'', Phys. Rev. E 70, No. 4, Article ID045101, 4 p. (2004; \url{doi:10.1103/PhysRevE.70.045101})]). The present paper studies how memory impacts passages at the origin for a ERW in the diffusive regime. It is shown that the number of zeros always grows asymptotically like the square root of the time. The problem is that, depending on the memory parameter, first return times to 0 may have a finite expectation or a fat tail with exponent less than 1/2. The author solves this problem by recasting the questions in the framework of scaling limits for self-similar Markov processes and for Markov chains.
Reviewer: Anatoliy Swishchuk (Calgary)Decay of harmonic functions for discrete time Feynman-Kac operators with confining potentialshttps://zbmath.org/1496.600872022-11-17T18:59:28.764376Z"Cygan, Wojciech"https://zbmath.org/authors/?q=ai:cygan.wojciech"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Śliwiński, Mateusz"https://zbmath.org/authors/?q=ai:sliwinski.mateuszSummary: We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in a countably infinite space. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.Shuffling cards by spatial motionhttps://zbmath.org/1496.600882022-11-17T18:59:28.764376Z"Diaconis, Persi"https://zbmath.org/authors/?q=ai:diaconis.persi-w"Pal, Soumik"https://zbmath.org/authors/?q=ai:pal.soumikSummary: We propose a model of card shuffling where a pack of cards, spread as points on a square table, are repeatedly gathered locally at random spots and then spread towards a random direction. A shuffling of the cards is then obtained by arranging the cards by their increasing \(x\)-coordinate values. When there are \(m\) cards on the table we show that this random ordering gets mixed in time \(O (\log m)\). Explicit constants are evaluated in a diffusion limit when the position of \(m\) cards evolves as an interesting \(2 m\)-dimensional non-reversible reflected jump diffusion in time. Our main technique involves the use of multidimensional Skorokhod maps for double reflections in \([0, 1]^2\) in taking the discrete to continuous limit. The limiting computations are then based on the planar Brownian motion and properties of Bessel processes.Explicit transient probabilities of various Markov modelshttps://zbmath.org/1496.600892022-11-17T18:59:28.764376Z"Krinik, Alan"https://zbmath.org/authors/?q=ai:krinik.alan-c"von Bremen, Hubertus"https://zbmath.org/authors/?q=ai:von-bremen.hubertus-f"Ventura, Ivan"https://zbmath.org/authors/?q=ai:ventura.ivan"Nguyen, Uyen Vietthanh"https://zbmath.org/authors/?q=ai:nguyen.uyen-vietthanh"Lin, Jeremy J."https://zbmath.org/authors/?q=ai:lin.jeremy-j"Lu, Thuy Vu Dieu"https://zbmath.org/authors/?q=ai:lu.thuy-vu-dieu"Luk, Chon In (Dave)"https://zbmath.org/authors/?q=ai:luk.chon-in"Yeh, Jeffrey"https://zbmath.org/authors/?q=ai:yeh.jeffrey"Cervantes, Luis A."https://zbmath.org/authors/?q=ai:cervantes.luis-a"Lyche, Samuel R."https://zbmath.org/authors/?q=ai:lyche.samuel-r"Marian, Brittney A."https://zbmath.org/authors/?q=ai:marian.brittney-a"Aljashamy, Saif A."https://zbmath.org/authors/?q=ai:aljashamy.saif-a"Dela, Mark"https://zbmath.org/authors/?q=ai:dela.mark"Oudich, Ali"https://zbmath.org/authors/?q=ai:oudich.ali"Ostadhassanpanjehali, Pedram"https://zbmath.org/authors/?q=ai:ostadhassanpanjehali.pedram"Phey, Lyheng"https://zbmath.org/authors/?q=ai:phey.lyheng"Perez, David"https://zbmath.org/authors/?q=ai:perez.david"Kath, John Joseph"https://zbmath.org/authors/?q=ai:kath.john-joseph"Demmin, Malachi C."https://zbmath.org/authors/?q=ai:demmin.malachi-c"Dawit, Yoseph"https://zbmath.org/authors/?q=ai:dawit.yoseph"Hoogendyk, Christine Carmen Marie"https://zbmath.org/authors/?q=ai:hoogendyk.christine-carmen-marie"Kim, Aaron"https://zbmath.org/authors/?q=ai:kim.aaron"McDonough, Matthew"https://zbmath.org/authors/?q=ai:mcdonough.matthew"Castillo, Adam Trevor"https://zbmath.org/authors/?q=ai:castillo.adam-trevor"Beecher, David"https://zbmath.org/authors/?q=ai:beecher.david"Wong, Weizhong"https://zbmath.org/authors/?q=ai:wong.weizhong"Ayeda, Heba"https://zbmath.org/authors/?q=ai:ayeda.hebaLet \(P\) be the transition probability matrix corresponding to a finite-state Markov chain. The paper under review deals with finding explicit eigenvalue formulas of \(P\) and the expressions of \(P^k\) where \(k=2, 3, 4\ldots\) for various Markov chains. These results are also applied to find probabilities of sample paths restricted to a strip and generalized ballot box problems.
For the entire collection see [Zbl 1493.60004].
Reviewer: Liping Li (Beijing)General methods for bounding multidimensional ruin probabilities in regime-switching modelshttps://zbmath.org/1496.600902022-11-17T18:59:28.764376Z"Gajek, Lesław"https://zbmath.org/authors/?q=ai:gajek.leslaw"Rudź, Marcin"https://zbmath.org/authors/?q=ai:rudz.marcinSummary: We present a universal methodology for bounding multidimensional ultimate ruin probabilities \(\boldsymbol{\Psi}\) in regime-switching models. Some new lower and upper bounds on \(\boldsymbol{\Psi}\) are given. The considered methods are applicable to several discrete- and continuous time risk models. As an example, we construct a variety of new two-sided operator bounds which converge to \(\boldsymbol{\Psi}\) with an exponential rate. Several numerical examples are also provided.An \(n\)-dimensional Rosenbrock distribution for Markov chain Monte Carlo testinghttps://zbmath.org/1496.600912022-11-17T18:59:28.764376Z"Pagani, Filippo"https://zbmath.org/authors/?q=ai:pagani.filippo"Wiegand, Martin"https://zbmath.org/authors/?q=ai:wiegand.martin"Nadarajah, Saralees"https://zbmath.org/authors/?q=ai:nadarajah.saraleesSummary: The Rosenbrock function is a ubiquitous benchmark problem in numerical optimization, and variants have been proposed to test the performance of Markov chain Monte Carlo algorithms on distributions with a curved and narrow shape. In this work we discuss the Rosenbrock distribution and the advantages and limitations of its current \(n\)-dimensional extensions. We then propose a new extension to arbitrary dimensions called the Hybrid Rosenbrock distribution, which addresses all the limitations that affect the current extensions. The Hybrid Rosenbrock distribution is composed of conditional normal kernels arranged in such a way that preserves the key features of the original Rosenbrock kernel. Moreover, due to its structure, the Hybrid Rosenbrock distribution is analytically tractable, and possesses several desirable properties which make it an excellent test model for computational algorithms. We conclude with numerical experiments that show how commonly used Markov chain Monte Carlo algorithms may fail to explore densities with curved correlation structure, restating the importance of a reliable benchmark problem for this class of densities.Regularity of models associated with Markov jump processeshttps://zbmath.org/1496.600922022-11-17T18:59:28.764376Z"Jedidi, Wissem"https://zbmath.org/authors/?q=ai:jedidi.wissem(no abstract)Estimates of heat kernels of non-symmetric Lévy processeshttps://zbmath.org/1496.600932022-11-17T18:59:28.764376Z"Grzywny, Tomasz"https://zbmath.org/authors/?q=ai:grzywny.tomasz"Szczypkowski, Karol"https://zbmath.org/authors/?q=ai:szczypkowski.karolThe authors investigate densities of vaguely continuous convolution semigroups of probability measures on \({\mathbb R}^d\). First, they provide the results that give the upper estimates in a situation in which the corresponding jump measure is allowed to be highly non-symmetric. Then, they provide upper estimates of the density and its derivatives, when the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. The authors also discuss the lower estimates, in view of a recent development in that direction, which complement the constructed upper estimates. Finally, the authors apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.
Reviewer: Pavel Gapeev (London)An averaging principle for stochastic flows and convergence of non-symmetric Dirichlet formshttps://zbmath.org/1496.600942022-11-17T18:59:28.764376Z"Barret, Florent"https://zbmath.org/authors/?q=ai:barret.florent"Raimond, Olivier"https://zbmath.org/authors/?q=ai:raimond.olivierLet \(X^\kappa\) be a diffusion process on a Riemannian manifold \(M\) with generator \(A^\kappa=A+\kappa V\), where \(A\) is a second order differential operator, \(V\) is a vector field on \(M\) and \(\kappa\) is a large positive parameter. After an appropriate time change, \(X^\kappa\) is a random perturbation of the dynamical system \(\frac{dx_t}{dt}=V(x_t)\). Suppose that one is able to construct a certain metric space \(\tilde{M}\) and a continuous map \(\pi: M\rightarrow \tilde{M}\). Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco convergence, the paper under review mainly obtains that under suitable conditions, \(\tilde{X}^\kappa:=\pi(X^\kappa)\) converges in law towards a diffusion process \(\tilde{X}\) on \(\tilde{M}\) as \(\kappa\rightarrow \infty\). This result permits higher dimensions than \(d=2\), and the case of dimension \(2\) was paid particular attention to in the book [\textit{M. I. Freidlin} and \textit{A. D. Wentzell}, Random perturbations of dynamical systems. Translated from the Russian by J Szücs. 3rd ed. Berlin: Springer (2012; Zbl 1267.60004)]. Furthermore, this result is also applied to a system of \(n\) particles in a turbulent fluid with a shear flow generated by \(V\).
Reviewer: Liping Li (Beijing)On local times of Ornstein-Uhlenbeck processeshttps://zbmath.org/1496.600952022-11-17T18:59:28.764376Z"Eisenbaum, Nathalie"https://zbmath.org/authors/?q=ai:eisenbaum.nathalieSummary: We establish expressions of the local time process of an Ornstein-Uhlenbeck process in terms of the local times on curves of a Brownian motion.
For the entire collection see [Zbl 1478.60005].Cooling down stochastic differential equations: Almost sure convergencehttps://zbmath.org/1496.600962022-11-17T18:59:28.764376Z"Dereich, Steffen"https://zbmath.org/authors/?q=ai:dereich.steffen"Kassing, Sebastian"https://zbmath.org/authors/?q=ai:kassing.sebastianSummary: We consider almost sure convergence of the SDE \(d X_t = \alpha_t d t + \beta_t d W_t\) under the existence of a \(C^2\)-Lyapunov function \(F : \mathbb{R}^d \to \mathbb{R} \). More explicitly, we show that on the event that the process stays local we have almost sure convergence in the Lyapunov function \(( F (X_t))_{t \geq 0}\) as well as \(\nabla F (X_t) \to 0\), if \(| \beta_t | = \mathcal{O} (t^{- \beta})\) for a \(\beta > 1 / 2\). If, additionally, one assumes that \(F\) is a Łojasiewicz-function, we get almost sure convergence of the process itself, given that \(| \beta_t | = \mathcal{O} (t^{- \beta})\) for a \(\beta > 1\). The assumptions are shown to be optimal in the sense that there is a divergent counterexample where \(|\beta_t|\) is of order \(t^{- 1} \).A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributionshttps://zbmath.org/1496.600972022-11-17T18:59:28.764376Z"Forman, Noah"https://zbmath.org/authors/?q=ai:forman.noah-mills"Rizzolo, Douglas"https://zbmath.org/authors/?q=ai:rizzolo.douglas"Shi, Quan"https://zbmath.org/authors/?q=ai:shi.quan"Winkel, Matthias"https://zbmath.org/authors/?q=ai:winkel.matthiasSummary: We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet \((\alpha,\theta)\) distributions, for \(\alpha \in (0,1)\) and \(\theta \geq 0\). These processes resolve a conjecture of \textit{S. Feng} and \textit{W. Sun} [Probab. Theory Relat. Fields 148, No. 3--4, 501--525 (2010; Zbl 1203.60120)]. We build on our previous work on \((\alpha,0)\)- and \((\alpha,\alpha)\)-interval partition evolutions. The extension to general \(\theta \geq 0\) is achieved by the construction of a \(\sigma\)-finite excursion measure of a new measure-valued branching diffusion. Our measure-valued processes are Hunt processes on an incomplete subspace of the space of all probability measures and do not possess an extension to a Feller process. In a companion paper, we use generators to show that ranked masses evolve according to a two-parameter family of diffusions introduced by \textit{L. A. Petrov} [Funct. Anal. Appl. 43, No. 4, 279--296 (2009; Zbl 1204.60076); translation from Funkts. Anal. Prilozh. 43, No. 4, 45--66 (2009)], extending work of \textit{S. N. Ethier} and \textit{T. G. Kurtz} [Adv. Appl. Probab. 13, 429--452 (1981; Zbl 0483.60076)].Heat kernel estimates on spaces with varying dimensionhttps://zbmath.org/1496.600982022-11-17T18:59:28.764376Z"Ooi, Takumu"https://zbmath.org/authors/?q=ai:ooi.takumuSummary: We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by \textit{Z.-Q. Chen} and \textit{S. Lou} [Ann. Probab. 47, No. 1, 213--269 (2019; Zbl 1466.60165)], in which a 1-dimensional space and a 2-dimensional space are connected at one point.Hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricinghttps://zbmath.org/1496.600992022-11-17T18:59:28.764376Z"Zhang, Haoyan"https://zbmath.org/authors/?q=ai:zhang.haoyan"Tian, Yingxu"https://zbmath.org/authors/?q=ai:tian.yingxuSummary: This paper investigates the hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricing. We study the Laplace transform of first hitting time over the constant and random jump boundary, respectively. The results about hitting the constant boundary serve for solving the optimal stopping problem of sticky Brownian motion. By introducing the sharpo ratio, we settle the bond pricing problem under sticky Brownian motion as well. An interesting result shows that the sticky point is in the continuation region and all the results we get are in closed form.Path decompositions of perturbed reflecting Brownian motionshttps://zbmath.org/1496.601002022-11-17T18:59:28.764376Z"Aïdékon, Elie"https://zbmath.org/authors/?q=ai:aidekon.elie-e-f"Hu, Yueyun"https://zbmath.org/authors/?q=ai:hu.yueyun"Shi, Zhan"https://zbmath.org/authors/?q=ai:shi.zhanSummary: We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by
\textit{G. F. Lawler} and \textit{W. Werner} [Probab. Theory Relat. Fields 128, No. 4, 565--588 (2004; Zbl 1049.60072)] and
\textit{Y. Le Jan} [Ann. Probab. 38, No. 3, 1280--1319 (2010; Zbl 1197.60075); Markov paths, loops and fields. École d'Été de Probabilités de Saint-Flour XXXVIII -- 2008. Berlin: Springer (2012; Zbl 1231.60002)],
in particular on a result discovered by
\textit{T. Lupu} [Mém. Soc. Math. Fr., Nouv. Sér. 158, 1--158 (2018; Zbl 1425.60004)] identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.
For the entire collection see [Zbl 1478.60005].Corrigendum and addendum to: ``Identification of the polaron measure. I: Fixed coupling regime and the central limit theorem for large times''https://zbmath.org/1496.601012022-11-17T18:59:28.764376Z"Mukherjee, Chiranjib"https://zbmath.org/authors/?q=ai:mukherjee.chiranjib"Varadhan, S. R. S."https://zbmath.org/authors/?q=ai:varadhan.s-r-srinivasaSummary: The proof of [the authors, ibid. 73, No. 2, 350--383 (2020; Zbl 1442.60082), Theorem 4.8] for large coupling constant \(\alpha\) has a gap (while the proof remains correct for small \(\alpha\)). We correct this error by giving a direct proof of [loc. cit., Theorem 4.8], which holds for any coupling parameter \(\alpha >0\). Consequently, the main results of [loc. cit.] on existence and identification of the Polaron measure and its CLT now hold for any \(\alpha >0\).Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion processhttps://zbmath.org/1496.601022022-11-17T18:59:28.764376Z"Abundo, Mario"https://zbmath.org/authors/?q=ai:abundo.marioSummary: We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If \(X(t)\) is a one-dimensional diffusion with jumps, starting from a random position \(\eta\in[a,b]\), let be \(\tau_{a,b}\) the time at which \(X(t)\) first exits the interval \((a,b)\), and \(\pi_a=P(X(\tau_{a,b})\le a)\) the probability of exit from the left of \((a,b)\). Given a probability \(q\in(0,1)\), the problem consists in finding the density \(g\) of \(\eta\) (if it exists) such that \(\pi_a=q\); it can be seen as a problem of optimization.Green function for gradient perturbation of unimodal Lévy processes in the real linehttps://zbmath.org/1496.601032022-11-17T18:59:28.764376Z"Grzywny, T."https://zbmath.org/authors/?q=ai:grzywny.tomasz"Jakubowski, T."https://zbmath.org/authors/?q=ai:jakubowski.tomasz"Żurek, G."https://zbmath.org/authors/?q=ai:zurek.grzegorzSummary: We prove that the Green function of the generator of symmetric unimodal Lévy process with the weak lower scaling order bigger than one and the Green functions of its gradient perturbations are comparable for bounded \(C^{1,1}\) subsets of the real line if the drift function is from an appropriate Kato class.Refined large deviation principle for branching Brownian motion conditioned to have a low maximumhttps://zbmath.org/1496.601042022-11-17T18:59:28.764376Z"Bai, Yanjia"https://zbmath.org/authors/?q=ai:bai.yanjia"Hartung, Lisa"https://zbmath.org/authors/?q=ai:hartung.lisa-barbelA binary branching Brownian motion is a particle system on the real line starting from a unique particle at position \(0\) at time \(0\), in which particle move according to independent Brownian motions, and split at rate \(1\) into two particles. Denote by \(\tau\) the first branching time, by \(y\) the position of the initial particle at that first branching time and by \(M_t\) the rightmost occupied position at time \(t\). It is well known that in this process, one has \(\lim_{t \to \infty} \frac{M_t}{t}=\sqrt{2}\) almost surely. The present article studies the behaviour of the branching Brownian motion conditioned on the large deviation event \(\{M_t<\sqrt{2}\alpha t\}\) for \(\alpha < 1\).
The large deviations for the maximal displacement of the branching Brownian motion were studied in [\textit{B. Derrida} and \textit{Z. Shi}, Springer Proc. Math. Stat. 208, 303--312 (2017; Zbl 1386.60290)], in which it is observed that conditioning the maximum to be small has the effect of suppressing the branching of the process for a positive proportion of the time and changing the speed of the initial particle. In the present article, the authors give precise estimates on the position and time of first branching of a branching Brownian motion conditioned on \(\{M_t<\sqrt{2}\alpha t\}\), obtaining a large deviations estimates for the couple \((\tau,y)\). The large deviations functional of this pair of variables exhibits a number of first and second order phase transitions.
Reviewer: Bastien Mallein (Paris)On a two-parameter Yule-Simon distributionhttps://zbmath.org/1496.601052022-11-17T18:59:28.764376Z"Baur, Erich"https://zbmath.org/authors/?q=ai:baur.erich"Bertoin, Jean"https://zbmath.org/authors/?q=ai:bertoin.jeanSummary: We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in
[the second author, J. Stat. Phys. 176, No. 3, 679--691 (2019; Zbl 1481.60184)]
in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in
[loc. cit.].
Finally, by superposing mutations to the branching process, we propose a model which leads to the two-parameter range of the Yule-Simon law, generalizing thereby the work of
\textit{H. A. Simon} [Biometrika 42, 425--440 (1955; Zbl 0066.11201)]
on limiting word frequencies.
For the entire collection see [Zbl 1478.60005].On the discounted penalty function in a perturbed Erlang renewal risk model with dependencehttps://zbmath.org/1496.601062022-11-17T18:59:28.764376Z"Adékambi, Franck"https://zbmath.org/authors/?q=ai:adekambi.franck"Takouda, Essodina"https://zbmath.org/authors/?q=ai:takouda.essodinaSummary: In this paper, we consider the risk model perturbed by a diffusion process. We assume an Erlang(n) risk process, \((n=1,2,\dots)\) to study the Gerber-Shiu discounted penalty function when ruin is due to claims or oscillations by including a dependence structure between claim sizes and their occurrence time. We derive the integro-differential equation of the expected discounted penalty function, its Laplace transform. Then, by analyzing the roots of the generalized Lundberg equation, we show that the expected penalty function satisfies a certain defective renewal equation and provide its representation solution. Finally, we give some explicit expressions for the Gerber-Shiu discounted penalty functions when the claim size distributions are Erlang(m), \((m=1,2,\dots)\) and provide numerical examples to illustrate the ruin probability.Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generationshttps://zbmath.org/1496.601072022-11-17T18:59:28.764376Z"Bohun, Vladyslav"https://zbmath.org/authors/?q=ai:bohun.vladyslav"Iksanov, Alexander"https://zbmath.org/authors/?q=ai:iksanov.aleksander-m"Marynych, Alexander"https://zbmath.org/authors/?q=ai:marynych.alexander-v"Rashytov, Bohdan"https://zbmath.org/authors/?q=ai:rashytov.bohdanSummary: An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem, and the key renewal theorem) for the number of \(j\)th-generation individuals with birth times \(\leq t\), when \(j,t\to\infty\) and \(j(t)=o\left(t^{2/3}\right)\). According to our terminology, such generations form a subset of the set of intermediate generations.Poisson generalized gamma process and its propertieshttps://zbmath.org/1496.601082022-11-17T18:59:28.764376Z"Hwan Cha, Ji"https://zbmath.org/authors/?q=ai:hwan-cha.ji"Mercier, Sophie"https://zbmath.org/authors/?q=ai:bloch-mercier.sophieSummary: Although the nonhomogeneous Poisson process has been intensively applied in practice, it has also its own limitations. In this paper, a new counting process model (called Poisson Generalized Gamma Process) is developed to overcome the limitations of the nonhomogeneous Poisson process. Initially, some basic stochastic properties are derived. It will be seen that this new counting process model includes both the generalized Pólya and Poisson Lindley processes as special cases. The influence of the model parameters on the behaviour of the new counting process model is analysed. The increments of the new process are shown to exhibit positive dependence properties. The corresponding compound process is defined and studied as well.Asymptotic sum-ruin probability for a bidimensional risk model with common shock dependencehttps://zbmath.org/1496.601092022-11-17T18:59:28.764376Z"Sun, Huimin"https://zbmath.org/authors/?q=ai:sun.huimin"Geng, Bingzhen"https://zbmath.org/authors/?q=ai:geng.bingzhen"Wang, Shijie"https://zbmath.org/authors/?q=ai:wang.shijieSummary: In this paper, we consider a bidimensional continuous-time renewal risk model with common shock claim-arrival processes. When both two lines of claim sizes are assumed to be strongly subexponential, a uniformly asymptotic formula for sum-ruin probability within finite time \(t\) is established. The obtained result extends the one in [\textit{D. Cheng} and \textit{C. Yu}, Commun. Stat., Theory Methods 49, No. 7, 1742--1760 (2020; Zbl 07528859)].A two-server bulk service queuing model with a permanent server and a temporary server on holdhttps://zbmath.org/1496.601102022-11-17T18:59:28.764376Z"Bakuli, Kuntal"https://zbmath.org/authors/?q=ai:bakuli.kuntal"Pal, Manisha"https://zbmath.org/authors/?q=ai:pal.manishaSummary: In this paper, we consider a bulk service queuing model with a fixed bulk size and a single permanent server. An additional server is kept on hold and is allowed to serve when the queue length exceeds certain threshold value. The model is analyzed using embedded Markov chain. A comparison of the performance of the model with the following models have also been made -- (i) two-server bulk service model, (ii) bulk service model with two independent queues corresponding to two servers and (iii) a single-server bulk service model with double service capacity of the server.Efficient steady-state simulation of high-dimensional stochastic networkshttps://zbmath.org/1496.601112022-11-17T18:59:28.764376Z"Blanchet, Jose"https://zbmath.org/authors/?q=ai:blanchet.jose-h"Chen, Xinyun"https://zbmath.org/authors/?q=ai:chen.xinyun"Si, Nian"https://zbmath.org/authors/?q=ai:si.nian"Glynn, Peter W."https://zbmath.org/authors/?q=ai:glynn.peter-wSummary: We propose and study an asymptotically optimal Monte Carlo estimator for steady-state expectations of a \(d\)-dimensional reflected Brownian motion (RBM). Our estimator is asymptotically optimal in the sense that it requires \(\tilde{O}(d)\) (up to logarithmic factors in \(d)\) independent and identically distributed scalar Gaussian random variables in order to output an estimate with a controlled error. Our construction is based on the analysis of a suitable multilevel Monte Carlo strategy which, we believe, can be applied widely. This is the first algorithm with linear complexity (under suitable regularity conditions) for a steady-state estimation of RBM as the dimension increases.The prelimit generator comparison approach of Stein's methodhttps://zbmath.org/1496.601122022-11-17T18:59:28.764376Z"Braverman, Anton"https://zbmath.org/authors/?q=ai:braverman.antonSummary: This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The ``standard'' generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the \textit{prelimit approach}. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the \(M / M / 1\) model as a simple working example to illustrate our approach.Approximations and optimal control for state-dependent limited processor sharing queueshttps://zbmath.org/1496.601132022-11-17T18:59:28.764376Z"Gupta, Varun"https://zbmath.org/authors/?q=ai:gupta.varun|gupta.varun.1|gupta.varun.2"Zhang, Jiheng"https://zbmath.org/authors/?q=ai:zhang.jihengAuthors' abstract: The paper studies approximations and control of a processor sharing (PS) server where the service rate depends on the number of jobs occupying the server. The control of such a system is implemented by imposing a limit on the number of jobs that can share the server concurrently, with the rest of the jobs waiting in a first-in-first-out (FIFO) buffer. A desirable control scheme should strike the right balance between efficiency (operating at a high service rate) and parallelism (preventing small jobs from getting stuck behind large ones). We use the framework of heavy-traffic diffusion analysis to devise near optimal control heuristics for such a queueing system. However, although the literature on diffusion control of state-dependent queueing systems begins with a sequence of systems and an exogenously defined drift function, we begin with a finite discrete PS server and propose an axiomatic recipe to explicitly construct a sequence of state-dependent PS servers that then yields a drift function. We establish diffusion approximations and use them to obtain insightful and closed-form approximations for the original system under a static concurrency limit control policy. We extend our study to control policies that dynamically adjust the concurrency limit. We provide two novel numerical algorithms to solve the associated diffusion control problem. Our algorithms can be viewed as ``average cost'' iteration: The first algorithm uses binary-search on the average cost, while the second faster algorithm uses Newton-Raphson method for root finding. Numerical experiments demonstrate the accuracy of our approximation for choosing optimal or near-optimal static and dynamic concurrency control heuristics.
Reviewer: Vyacheslav Abramov (Melbourne)Queueing models for cognitive wireless networks with sensing time of secondary usershttps://zbmath.org/1496.601142022-11-17T18:59:28.764376Z"Phung-Duc, Tuan"https://zbmath.org/authors/?q=ai:phung-duc.tuan"Akutsu, Kohei"https://zbmath.org/authors/?q=ai:akutsu.kohei"Kawanishi, Ken'ichi"https://zbmath.org/authors/?q=ai:kawanishi.kenichi"Salameh, Osama"https://zbmath.org/authors/?q=ai:salameh.osama"Wittevrongel, Sabine"https://zbmath.org/authors/?q=ai:wittevrongel.sabineSummary: This paper considers queueing models for cognitive radio networks that account for the sensing time of secondary users (SUs). In cognitive radio networks, secondary users are allowed to opportunistically use idle channels originally allocated to primary users (PUs). To this end, SUs must sense the state of the channels before transmission. After sensing, if an idle channel is available, the SU can start transmission immediately; otherwise, the SU must carry out another channel sensing. In this paper, we study two retrial queueing models with an unlimited number of sensing SUs, where PUs have preemptive priority over SUs. The two models differ in whether or not an arriving PU can interrupt a SU transmission also in case there are still idle channels available. We show that both models have the same stability condition and that the model without interruptions in case of available idle channels has a slightly lower number of sensing SUs than the model with interruptions.Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscapehttps://zbmath.org/1496.601152022-11-17T18:59:28.764376Z"Bates, Erik"https://zbmath.org/authors/?q=ai:bates.erik"Ganguly, Shirshendu"https://zbmath.org/authors/?q=ai:ganguly.shirshendu"Hammond, Alan"https://zbmath.org/authors/?q=ai:hammond.alanSummary: Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [``The directed landscape'', Preprint, \url{arXiv:1812.00309}] of \textit{D. Dauvergne} et al., this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points \((x,s)\) and \((y,t)\) with \(s< t\). In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations \(x_1\) and \(x_2\), and consider geodesics traveling \((x_1, 0)\to (y,1)\) and \((x_2, 0)\to (y,1)\). We prove that the set of \(y\in \mathbb{R}\) for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints \((x,0)\) and \((y,1)\) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such \((x,y)\in\mathbb{R}^2\) also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called \textit{difference weight profile} studied in
[\textit{R. Basu} et al., Ann. Probab. 49, No. 1, 485--505 (2021; Zbl 1457.82165)]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [\textit{A. Hammond}, Proc. Lond. Math. Soc. (3) 120, No. 3, 370--433 (2020; Zbl 1453.82078)].Chase-escape on the configuration modelhttps://zbmath.org/1496.601162022-11-17T18:59:28.764376Z"Bernstein, Emma"https://zbmath.org/authors/?q=ai:bernstein.emma"Hamblen, Clare"https://zbmath.org/authors/?q=ai:hamblen.clare"Junge, Matthew"https://zbmath.org/authors/?q=ai:junge.matthew"Reeves, Lily"https://zbmath.org/authors/?q=ai:reeves.lilySummary: Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-\(\lambda\) Poisson process while being chased and consumed by blue particles according to a rate-\(1\) Poisson process. Given a growing sequence of finite graphs, the critical rate \(\lambda_c\) is the largest value of \(\lambda\) for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on \(\lambda_c\) for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactionshttps://zbmath.org/1496.601172022-11-17T18:59:28.764376Z"Houdebert, Pierre"https://zbmath.org/authors/?q=ai:houdebert.pierre"Zass, Alexander"https://zbmath.org/authors/?q=ai:zass.alexanderSummary: We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity \(z\) and inverse temperature \(\beta\). The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.The logarithmic anti-derivative of the baik-rains distribution satisfies the KP equationhttps://zbmath.org/1496.601182022-11-17T18:59:28.764376Z"Zhang, Xincheng"https://zbmath.org/authors/?q=ai:zhang.xinchengSummary: It has been discovered that the Kadomtsev-Petviashvili (KP) equation governs the distribution of the fluctuation of many random growth models. In particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of the Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. The result is first derived formally by taking a limit of the generating function of the KPZ equation, which satisfies the KP equation. Then we prove it directly using the explicit Painlevé II formulation of the Baik-Rains distribution.On Sobolev rough pathshttps://zbmath.org/1496.601192022-11-17T18:59:28.764376Z"Liu, Chong"https://zbmath.org/authors/?q=ai:liu.chong"Prömel, David J."https://zbmath.org/authors/?q=ai:promel.david-j"Teichmann, Josef"https://zbmath.org/authors/?q=ai:teichmann.josefThe authors consider spaces of rough paths with Sobolev regularity and associated controlled rough differential equations. First they define the space of Sobolev rough paths using a fractional Sobolev norm. Typical estimates used for rough paths work for \(p\)-variation or related seminorms, but not for the fractional Sobolev norm presenting the main obstacle. In a suitable topology, the authors introduce the controlled rough paths of Sobolev type requiring a remainder term in the mixed Hölder-variation space. Then they show that solutions to the controlled rough differential equations driven by Sobolev rough paths also possess Sobolev regularity. Finally the authors prove local Lipschitz continuity of the Itô-Lyons map on the space of Sobolev rough paths with arbitrary low regularity with respect to the initial value, vector field and the driving signal.
Reviewer: Maria Gordina (Storrs)Split Hamiltonian Monte Carlo revisitedhttps://zbmath.org/1496.620062022-11-17T18:59:28.764376Z"Casas, Fernando"https://zbmath.org/authors/?q=ai:casas.fernando"Sanz-Serna, Jesús María"https://zbmath.org/authors/?q=ai:sanz-serna.jesus-maria"Shaw, Luke"https://zbmath.org/authors/?q=ai:shaw.lukeSummary: We study Hamiltonian Monte Carlo (HMC) samplers based on splitting the Hamiltonian \(H\) as \(H_0(\theta ,p)+U_1(\theta )\), where \(H_0\) is quadratic and \(U_1\) small. We show that, in general, such samplers suffer from stepsize stability restrictions similar to those of algorithms based on the standard leapfrog integrator. The restrictions may be circumvented by preconditioning the dynamics. Numerical experiments show that, when the \(H_0(\theta ,p)+U_1(\theta )\) splitting is combined with preconditioning, it is possible to construct samplers far more efficient than standard leapfrog HMC.Mixture of multivariate Gaussian processes for classification of irregularly sampled satellite image time-serieshttps://zbmath.org/1496.620092022-11-17T18:59:28.764376Z"Constantin, Alexandre"https://zbmath.org/authors/?q=ai:constantin.alexandre"Fauvel, Mathieu"https://zbmath.org/authors/?q=ai:fauvel.mathieu"Girard, Stéphane"https://zbmath.org/authors/?q=ai:girard.stephaneSummary: The classification of irregularly sampled Satellite image time-series (SITS) is investigated in this paper. A multivariate Gaussian process mixture model is proposed to address the irregular sampling, the multivariate nature of the time-series and the scalability to large data-sets. The spectral and temporal correlation is handled using a Kronecker structure on the covariance operator of the Gaussian process. The multivariate Gaussian process mixture model allows both for the classification of time-series and the imputation of missing values. Experimental results on simulated and real SITS data illustrate the importance of taking into account the spectral correlation to ensure a good behavior in terms of classification accuracy and reconstruction errors.Efficient reduced-rank methods for Gaussian processes with eigenfunction expansionshttps://zbmath.org/1496.620122022-11-17T18:59:28.764376Z"Greengard, Philip"https://zbmath.org/authors/?q=ai:greengard.philip"O'Neil, Michael"https://zbmath.org/authors/?q=ai:oneil.michaelSummary: In this work, we introduce a reduced-rank algorithm for Gaussian process regression. Our numerical scheme converts a Gaussian process on a user-specified interval to its Karhunen-Loève expansion, the \(L^2\)-optimal reduced-rank representation. Numerical evaluation of the Karhunen-Loève expansion is performed once during precomputation and involves computing a numerical eigendecomposition of an integral operator whose kernel is the covariance function of the Gaussian process. The Karhunen-Loève expansion is independent of observed data and depends only on the covariance kernel and the size of the interval on which the Gaussian process is defined. The scheme of this paper does not require translation invariance of the covariance kernel. We also introduce a class of fast algorithms for Bayesian fitting of hyperparameters and demonstrate the performance of our algorithms with numerical experiments in one and two dimensions. Extensions to higher dimensions are mathematically straightforward but suffer from the standard curses of high dimensions.Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamicshttps://zbmath.org/1496.620252022-11-17T18:59:28.764376Z"Zhang, Benjamin J."https://zbmath.org/authors/?q=ai:zhang.benjamin-j"Marzouk, Youssef M."https://zbmath.org/authors/?q=ai:marzouk.youssef-m"Spiliopoulos, Konstantinos"https://zbmath.org/authors/?q=ai:spiliopoulos.konstantinos-vSummary: We introduce a novel geometry-informed irreversible perturbation that accelerates convergence of the Langevin algorithm for Bayesian computation. It is well documented that there exist perturbations to the Langevin dynamics that preserve its invariant measure while accelerating its convergence. Irreversible perturbations and reversible perturbations (such as Riemannian manifold Langevin dynamics (RMLD)) have separately been shown to improve the performance of Langevin samplers. We consider these two perturbations simultaneously by presenting a novel form of irreversible perturbation for RMLD that is informed by the underlying geometry. Through numerical examples, we show that this new irreversible perturbation can improve estimation performance over irreversible perturbations that do not take the geometry into account. Moreover we demonstrate that irreversible perturbations generally can be implemented in conjunction with the stochastic gradient version of the Langevin algorithm. Lastly, while continuous-time irreversible perturbations cannot impair the performance of a Langevin estimator, the situation can sometimes be more complicated when discretization is considered. To this end, we describe a discrete-time example in which irreversibility increases both the bias and variance of the resulting estimator.Improper priors and improper posteriorshttps://zbmath.org/1496.620272022-11-17T18:59:28.764376Z"Taraldsen, Gunnar"https://zbmath.org/authors/?q=ai:taraldsen.gunnar"Tufto, Jarle"https://zbmath.org/authors/?q=ai:tufto.jarle"Lindqvist, Bo H."https://zbmath.org/authors/?q=ai:lindqvist.bo-henrySummary: What is a good prior? Actual prior knowledge should be used, but for complex models this is often not easily available. The knowledge can be in the form of symmetry assumptions, and then the choice will typically be an improper prior. Also more generally, it is quite common to choose improper priors. Motivated by this we consider a theoretical framework for statistics that includes both improper priors and improper posteriors. Knowledge is then represented by a possibly unbounded measure with interpretation as explained by Rényi in [Acta Math. Acad. Sci. Hung. 6, 285--335 (1955; Zbl 0067.10401)]. The main mathematical result here is a constructive proof of existence of a transformation from prior to posterior knowledge. The posterior always exists and is uniquely defined by the prior, the observed data, and the statistical model. The transformation is, as it should be, an extension of conventional Bayesian inference as defined by the axioms of Kolmogorov. It is an extension since the novel construction is valid also when replacing the axioms of Kolmogorov by the axioms of Rényi for a conditional probability space. A concrete case based on Markov Chain Monte Carlo simulations and data for different species of tropical butterflies illustrate that an improper posterior may appear naturally and is useful. The theory is also exemplified by more elementary examples.Alpha power transformation of Lomax distribution: properties and applicationshttps://zbmath.org/1496.620432022-11-17T18:59:28.764376Z"Maruthan, Sakthivel Kandaswamy"https://zbmath.org/authors/?q=ai:maruthan.sakthivel-kandaswamy"Venkatachalam, Nandhini"https://zbmath.org/authors/?q=ai:venkatachalam.nandhiniSummary: In this paper, we present a new three-parameter alpha power transformation of Lomax distribution (APTLx). Some statistical properties of the APTLx distribution are obtained including moments, quantiles, entropy, order statistics, and stress-strength analysis and its explicit expressions are derived. Maximum likelihood estimation method is used to estimate the parameters of the distribution. The goodness-of-fit of the proposed model show that the new distribution performs favorably when compare with existing distributions. The application of APTLx distribution is emphasized using a real-life data.A novel Lomax extension with statistical properties, copulas, different estimation methods and applicationshttps://zbmath.org/1496.620472022-11-17T18:59:28.764376Z"Aboraya, Mohamed"https://zbmath.org/authors/?q=ai:aboraya.mohamed"Ali, M. Masoom"https://zbmath.org/authors/?q=ai:ali.mir-masoom.1|masoom-ali.m"Yousof, Haitham M."https://zbmath.org/authors/?q=ai:yousof.haitham-mosaad|yousof.haitham-mosad"Ibrahim, Mohamed"https://zbmath.org/authors/?q=ai:ibrahim.mohamed-salah|ibrahim.mohamed-hamdi|ibrahim.mohamed-hamza|ibrahim.mohamed-aSummary: A new compound Lomax model is proposed and analyzed. The novel distribution is derived based on compounding the zero truncated Poisson distribution and the exponentiated exponential Lomax distribution. The new density can be ``monotonically left skewed'', ``monotonically right skewed'' and ``symmetric'' with various useful shapes. The new hazard rate can be ``upside down bathtub-increasing'', ``bathtub (\textbf{U}-shape)'', ``monotonically decreasing'', ``increasing-constant'' and ``monotonically increasing''. Relevant statistical properties are derived. We briefly describe different estimation methods, namely the maximum likelihood, Cramér-von-Mises, ordinary least squares, weighted least square, Anderson-Darling, right tail Anderson-Darling and left tail Anderson-Darling. Monte Carlo simulation experiments are performed for comparing the performances of the proposed methods of estimation for both small and large samples. For facilitating the mathematical modeling of the bivariate real data sets, we derive some new corresponding bivariate distributions. Graphical simulation study is performed for assessing the finite sample behavior of the estimators using the maximum likelihood method. Two applications are provided for illustrating the applicability of the new model.A new flexible three-parameter compound Chen distribution: properties, copula and modeling relief times and minimum flow datahttps://zbmath.org/1496.620482022-11-17T18:59:28.764376Z"Ali, M. Masoom"https://zbmath.org/authors/?q=ai:masoom-ali.m|ali.mir-masoom.1"Ibrahim, Mohamed"https://zbmath.org/authors/?q=ai:ibrahim.mohamed-salah|ibrahim.mohamed-hamdi|ibrahim.mohamed-hamza|ibrahim.mohamed-a"Yousof, Haitham M."https://zbmath.org/authors/?q=ai:yousof.haitham-mosad|yousof.haitham-mosaadSummary: In this work, a new flexible extension of the Chen distribution is derived and studied. The new density accommodates the ``unimodal right skewed'', ``unimodal left skewed'', ``bimodal right skewed'' and ``bimodal left skewed'' shapes. The new hazard rate can be ``decreasing-constant-increasing (bathtub)'', ``monotonically increasing'', ``upside down constant-increasing'', ``monotonically decreasing'', ``upside down'' and ``\textbf{J} shape''. Some of its statistical properties such as moments, mean residual lifetime, conditional moments and mean past lifetime are derived. Some bivariate extensions using Ali-Mikhail-Haq copula, Renyi copula, Farlie-Gumbel-Morgenstern, Clayton copula and modified Farlie-Gumbel-Morgenstern copula are derived. Two real data sets are analyzed for illustrating the wide applicability of the new model. The new model is compared with many common competitive models under some well-known goodness-of-fit statistic criteria.A new compound exponentiated Weibull distribution with statistical properties and applicationshttps://zbmath.org/1496.620492022-11-17T18:59:28.764376Z"Ansari, Saiful Islam"https://zbmath.org/authors/?q=ai:ansari.saiful-islamSummary: This paper is intended to design a new distribution focused on Poisson zero truncated that also accommodates many significant failure rates. Several of its characteristics are derived mathematically. The actual model's density is being interpreted as a mixture of exponentiated Weibull densities. The maximum likelihood method is considered to estimate model parameters. We demonstrated the significance and versatility of the compound exponentiated Weibull distribution through simulation of three data sets.The alpha power shifted exponential distribution: properties and applicationshttps://zbmath.org/1496.620502022-11-17T18:59:28.764376Z"Eghwerido, Joseph Thomas"https://zbmath.org/authors/?q=ai:eghwerido.joseph-thomas"Agu, Ikechukwu Friday"https://zbmath.org/authors/?q=ai:agu.ikechukwu-fridaySummary: This article proposes a new three parameter distribution in the family of the exponential distribution called the alpha power shifted exponential (APOSE) distribution. The structural mathematical properties of the APOSE model were derived and examined. The APOSE model parameters were established and obtained by maximum likelihood method. The flexibility, efficiency, and behavior of the APOSE model estimators were examined. The results show that the APOSE distribution is increasing, decreasing, unimodal, and right skewed property. The bivariate model of the APOSE distribution was also proposed. The empirical applicability and proficiency of the APOSE model was examined by a real-life dataset. The empirical results show that the proposed APOSE model provides a better goodness-of-fit when compared to existing models in statistical literature and can serve as an alternative model to those appearing in modeling Poisson processes.On the exact distribution of the difference between two chi-square variableshttps://zbmath.org/1496.620522022-11-17T18:59:28.764376Z"Joarder, Anwar H."https://zbmath.org/authors/?q=ai:joarder.anwar-h"Omar, M. Hafidz"https://zbmath.org/authors/?q=ai:omar.m-hafidzSummary: We have reviewed the development of the exact distribution of the difference of two independently and identically distributed chi-square variables and presented derivations based on moment generating function and also by convolution method. We have also developed general moment results of the difference of two independently and identically distributed random variables and applied these to chi-square random variables. Some properties of the distribution, namely moment generating function, mean centered moments, coefficient of skewness and kurtosis, and cumulative distribution function, have been derived. The graph of the density functions has been presented. We conclude the paper with directions for further research along the line.A new zero-one-inflated Poisson-Lindley distribution for modelling overdispersed count datahttps://zbmath.org/1496.620542022-11-17T18:59:28.764376Z"Tajuddin, Razik Ridzuan Mohd"https://zbmath.org/authors/?q=ai:tajuddin.razik-ridzuan-mohd"Ismail, Noriszura"https://zbmath.org/authors/?q=ai:ismail.noriszura"Ibrahim, Kamarulzaman"https://zbmath.org/authors/?q=ai:ibrahim.kamarulzaman"Bakar, Shaiful Anuar Abu"https://zbmath.org/authors/?q=ai:bakar.shaiful-anuar-abuSummary: Many studies have considered mixed Poisson distributions as alternatives for fitting count data with overdispersion. However, in some cases, the data have an abundance of zeros and ones which makes modelling using distributions with no consideration to the inflated values less desirable. This study aims to introduce a new distribution for count data with inflated values at zero and one, known as zero-one-inflated Poisson-Lindley distribution. The statistical properties of the proposed distribution are discussed. Furthermore, the maximum likelihood and method of moments for the parameters of the proposed distribution are developed. A simulation study is conducted to investigate the performance of the zero-one-inflated Poisson-Lindley distribution in describing overdispersed data with excess zeros and ones by changing the proportion of zeros and ones in the data. It is found that the fitting of the zero-one-inflated Poisson-Lindley distribution always gives a larger log-likelihood value than the fitting of the zero-one-inflated Poisson distribution. The results from the applications of the real datasets with an overdispersed property as well as a large number of ones and zeros conclude that the proposed distribution provides the best fit compared to other contending distributions in the study.The beta Topp-Leone generated family of distributions and theirs applicationshttps://zbmath.org/1496.620552022-11-17T18:59:28.764376Z"Watthanawisut, Atchariya"https://zbmath.org/authors/?q=ai:watthanawisut.atchariya"Bodhisuwan, Winai"https://zbmath.org/authors/?q=ai:bodhisuwan.winai"Supapakorn, Thidaporn"https://zbmath.org/authors/?q=ai:supapakorn.thidapornSummary: This work aims to establish a new family of distributions, namely the beta Topp-Leone generated family of distributions. The proposed family of distributions is combined from two families: the beta generated family and the Topp-Leone generated family. Some statistical properties of the proposed family are derived, e.g., linear representation, ordinary moments, and moment generating function. Furthermore, a new modification of Weibull distribution, namely the beta Topp-Leone Weibull distribution, is studied. The beta Topp-Leone Weibull distribution has flexible hazard shapes. Some statistical properties of the proposed distribution are studied, e.g., transformation, quantile function, ordinary moments, and moment generating function. The distribution parameters are estimated with the methods of maximum likelihood estimation. The proposed distribution shows more appropriate than other candidate distributions for fitting with the complete and censored datasets based on the values of Akaike's information criterion, Bayesian information criterion, Akaike's information corrected criterion, and Hannon and Quinn's information criterion.Asymptotic approximations for some distributions of ratioshttps://zbmath.org/1496.620562022-11-17T18:59:28.764376Z"Joutard, Cyrille"https://zbmath.org/authors/?q=ai:joutard.cyrilleSummary: We give strong large deviation results for some ratio distributions. Then we apply these results to two statistical examples: a ratio distribution with sums of gamma-distributed random variables and another one with sums of \(\chi^2\)-distributed random variables. We eventually carry out numerical comparisons with a saddlepoint approximation using an indirect Edgeworth expansion and a Lugannani and Rice saddlepoint approximation.Modeling and simulation studies for some truncated discrete distributions generated by stable densitieshttps://zbmath.org/1496.620572022-11-17T18:59:28.764376Z"Farbod, Davood"https://zbmath.org/authors/?q=ai:farbod.davoodSummary: Some discrete distributions generated by stable densities (DGSDs) could be considered as models for describing phenomena arising in bioinformatics. Since probability mass and distribution functions are not in closed forms, simulation studies and real applications of these distributions have not been studied yet. To do this, we need to consider DGSDs as truncated, namely, truncated DGSDs (T-DGSDs). In this paper, some statistical properties of the T-DGSDs models are established. Based on the Monte Carlo method, limited-memory Broyden-Fletcher-Goldfarb-Shanno for bound-constrained optimization, and Nelder-Mead optimization algorithms, we do a simulation to estimate biases, mean square errors, and maximum likelihood estimations for the unknown parameters of the T-DGSDs. Moreover, we fit these T-DGSDs models with some real data sets in bioinformatics and then compare them to some frequency distributions.Bayesian non-parametric priors based on random setshttps://zbmath.org/1496.620662022-11-17T18:59:28.764376Z"Gil-Leyva, María F."https://zbmath.org/authors/?q=ai:gil-leyva.maria-fSummary: We study the construction of random discrete distributions, taking values in the infinite dimensional simplex, by means of a latent random subset of the natural numbers. The derived sequences of random weights are then used to establish a Bayesian non-parametric prior. A sufficient condition on the distribution of the random set is given, that assures the corresponding prior has full support, and taking advantage of the construction, we propose a general MCMC algorithm for density estimation purposes. This method is illustrated by building a new distribution over the space of all finite and non-empty subsets of \(\mathbb{N}\), that subsequently leads to a general class of random probability measures termed Geometric product stick-breaking process. It is shown that Geometric product stick-breaking process approximate, in distribution, Dirichlet and Geometric processes, and that the respective weights sequences have heavy tails, thus leading to very flexible mixture models.
For the entire collection see [Zbl 1478.60006].A new family of Archimedean copulas: the truncated-Poisson family of copulashttps://zbmath.org/1496.620882022-11-17T18:59:28.764376Z"Alzaid, Abdulhamid A."https://zbmath.org/authors/?q=ai:alzaid.abdulhamid-a"Alhadlaq, Weaam M."https://zbmath.org/authors/?q=ai:alhadlaq.weaam-mSummary: Copulas are multivariate distribution functions which their margins are distributed uniformly. Therefore, copulas are pretty useful for modeling several types of data. As they allow different dependence patterns. A numerous number of new classes of copulas have been suggested in the literature. Each granted different characteristics that make it compatible with certain type of data. In this paper, we introduce a new family of Archimedean copulas. The multiplicative Archimedean generator of this copula is the inverse of the probability generating function of a truncated-Poisson distribution. The properties of this copula are studied in detail. Three applications are provided for the sake of comparison between this copula and well-known ones.Corrigendum to: ``On the unification of families of skew-normal distributions''https://zbmath.org/1496.620892022-11-17T18:59:28.764376Z"Arellano-Valle, Reinaldo B."https://zbmath.org/authors/?q=ai:arellano-valle.reinaldo-boris"Azzalini, Adelchi"https://zbmath.org/authors/?q=ai:azzalini.adelchiCorrigendum to the authors' paper [ibid. 33, No. 3, 561--574 (2006; Zbl 1117.62051)].Transport distances on random vectors of measures: recent advances in Bayesian nonparametricshttps://zbmath.org/1496.620902022-11-17T18:59:28.764376Z"Catalano, Marta"https://zbmath.org/authors/?q=ai:catalano.marta"Lijoi, Antonio"https://zbmath.org/authors/?q=ai:lijoi.antonio"Prünster, Igor"https://zbmath.org/authors/?q=ai:prunster.igorSummary: Random vectors of measures are at the core of many recent developments in Bayesian nonparametrics. For a deep understanding of these infinite-dimensional discrete random structures and their impact on the inferential and theoretical properties of the induced models, we consider a class of transport distances based on the Wasserstein distance. The geometrical definition makes it ideal for measuring similarity between distributions with possibly different supports. Moreover, when applied to random vectors of measures with independent increments (\textit{completely random vectors}), the interesting theoretical properties are coupled with analytical tractability. This leads to a new measure of dependence for completely random vectors and the quantification of the impact of hyperparameters in notable models for exchangeable time-to-event data.
For the entire collection see [Zbl 1478.60006].Finite mixtures of multivariate skew Student's \(t\) distributions with independent logistic skewing functionshttps://zbmath.org/1496.620912022-11-17T18:59:28.764376Z"Kwong, Hok Shing"https://zbmath.org/authors/?q=ai:kwong.hok-shing"Nadarajah, Saralees"https://zbmath.org/authors/?q=ai:nadarajah.saraleesSummary: This paper extends the multivariate skew \(t\) distributions with independent logistic skewing functions (MSTIL) introduced in [\textit{H. S. Kwong} and \textit{S. Nadarajah}, Methodol. Comput. Appl. Probab. 24, No. 3, 1669--1691 (2022; Zbl 1491.62040)] to finite mixture models (FM-MSTIL). A stochastic EM-type algorithm is proposed for fitting the FM-MSTIL, and a divisive hierarchical algorithm is proposed for initialisations and model selections. We show that the model can outperform other finite mixture models in the literature for some simulated data sets. The performance of the FM-MSTIL in cluster analysis is also investigated. We show that the FM-MSTIL-R, a nested version of the FM-MSTIL, performs well for automatic gating tasks on some flow cytometry data sets in the FlowCap-I challenge. The FM-MSTIL-R achieved a better overall score than all other competing algorithms in the original challenge. An efficient implementation of the FM-MSTIL is available as an R package in GitHub.Tractable circula densities from Fourier serieshttps://zbmath.org/1496.620932022-11-17T18:59:28.764376Z"Kato, Shogo"https://zbmath.org/authors/?q=ai:kato.shogo"Pewsey, Arthur"https://zbmath.org/authors/?q=ai:pewsey.arthur"Jones, M. C."https://zbmath.org/authors/?q=ai:jones.michael-chrisSummary: This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined `circulas'. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions.Circular generalized logistic distributions and its applicationshttps://zbmath.org/1496.620942022-11-17T18:59:28.764376Z"Maruthan, Sakthivel Kandasamy"https://zbmath.org/authors/?q=ai:maruthan.sakthivel-kandasamy"Bava, Alshad Karippayil"https://zbmath.org/authors/?q=ai:bava.alshad-karippayilSummary: This paper introduces three circular distributions based on inverse stereographic projection by considering three types of logistic distribution available in the literature, namely Type-I generalized logistic distribution, three parameter Type-I generalized logistic distribution and Type-II generalized logistic distribution. The proposed new models are more flexible than stereographic logistic model [\textit{A. V. Dattatreya Rao} et al., Chil. J. Stat. 7, No. 2, 69--79 (2016; Zbl 1463.62155)] to the optimal modeling of circular data, can take skewness and heavy tails into account and offer a great applicability in many fields. Since one of the proposed models called circular Type-II generalized logistic distribution shows more applicability than the other proposed models. Hence, in this paper we discuss the method maximum likelihood estimation and a simulation study for circular type-II generalized logistic distribution. Finally, four circular data-sets are considered for illustrative purposes and proving that the proposed models perform well than several other circular distributions.Estimations of means and variances in a Markov linear modelhttps://zbmath.org/1496.620952022-11-17T18:59:28.764376Z"Gutierrez, Abraham"https://zbmath.org/authors/?q=ai:gutierrez.abraham"Müller, Sebastian"https://zbmath.org/authors/?q=ai:muller.sebastian.4|muller.sebastian.3|muller.sebastian-b|muller.sebastian.1|muller.sebastian.2Summary: Multivariate regression models and ANOVA are probably the most frequently applied methods of all statistical analyses. We study the case where the predictors are qualitative variables and the response variable is quantitative. In this case, we propose an alternative to the classic approaches that does not assume homoscedasticity but assumes that a Markov chain can describe the covariates' correlations. This approach transforms the dependent covariates using a change of measure to independent covariates. The transformed estimates allow a pairwise comparison of the mean and variance of the contribution of different values of the covariates. We show that, under standard moment conditions, the estimators are asymptotically normally distributed. We test our method with data from simulations and apply it to several classic data sets.Tests of multivariate copula exchangeability based on Lévy measureshttps://zbmath.org/1496.620992022-11-17T18:59:28.764376Z"Bahraoui, Tarik"https://zbmath.org/authors/?q=ai:bahraoui.tarik"Quessy, Jean-François"https://zbmath.org/authors/?q=ai:quessy.jean-francoisSummary: This paper introduces tests for the symmetry of the copula of random vector. The proposed statistics are based on the copula characteristic function and the weight function that appears naturally in their definition are assumed to belong to the general family of Lévy measures. The proposed test statistics are rank-based and expresses as weighted \(L_2\)-norms computed from a vector of empirical copula characteristic functions. Their nondegenerate asymptotic distributions under the null hypothesis and general alternatives, as well as the validity of a multiplier bootstrap for the computation of \(p\)-values, are derived using nonstandard arguments. Extended Monte-Carlo experiments show that the new tests hold their size well and are powerful against a wide range of alternatives, and appear to be more powerful than a Cramér-von Mises test based on empirical copulas.High-dimensional sphericity test by extended likelihood ratiohttps://zbmath.org/1496.621002022-11-17T18:59:28.764376Z"Wang, Zhendong"https://zbmath.org/authors/?q=ai:wang.zhendong"Xu, Xingzhong"https://zbmath.org/authors/?q=ai:xu.xingzhongThe paper puts forward a novel sphericity test for covariance matrices by means of the likelihood ratio test. The advantage of this test lies in its ability to work even for high-dimensional data. The theoretical properties are proved and Monte Carlo simulations demonstrate that it overpasses state-of-the-art statistical tests in terms of empirical power.
Reviewer: Ruxandra Stoean (Craiova)On some properties of \(\alpha\)-mixtureshttps://zbmath.org/1496.621082022-11-17T18:59:28.764376Z"Shojaee, Omid"https://zbmath.org/authors/?q=ai:shojaee.omid"Asadi, Majid"https://zbmath.org/authors/?q=ai:asadi.majid"Finkelstein, Maxim"https://zbmath.org/authors/?q=ai:finkelstein.maximThe paper studies several properties of \(\alpha\)-mixtures for survival functions. The theoretical demonstrations refer to the ageing properties for additive and multiplicative hazards, the partial stochastic and hazard rate orderings of the finite \(\alpha\)-mixtures and the extension of bending properties to this type of mixtures.
Reviewer: Catalin Stoean (Craiova)Generalized linear model for subordinated Lévy processeshttps://zbmath.org/1496.621272022-11-17T18:59:28.764376Z"Mselmi, Farouk"https://zbmath.org/authors/?q=ai:mselmi.faroukSummary: Generalized linear models, introduced by Nelder and Wedderburn, allowed to model the regression of normal and nonnormal data. While doing so, the analysis of these models could not be obtained without the explicit form of the variance function. In this paper, we determine the link and variance functions of the natural exponential family generated by the class of subordinated Lévy processes. In this framework, we introduce a class of variance functions that depends on the Lambert function. In this regard, we call it the Lambert class, which covers the variance functions of the natural exponential families generated by the subordinated gamma processes and the subordinated Lévy processes by the Poisson subordinator. Notice that the gamma process subordinated by the Poisson one is excluded from this class. The concept of reciprocity in natural exponential families was given in order to obtain an exponential family from another one. In this context, we get the reciprocal class of the natural exponential family generated by the class of subordinated Lévy processes. It is well known that the variance function represents an essential element for the determination of the quasi-likelihood and deviance functions. Then, we use the expression of our variance function in order to maintain them. This leads us to analyze the proposed generalized linear model. We illustrate some of our models with applications to the daily exchange rate returns of the Tunisian Dinar against the U.S. Dollar and the damage incidents of ships.Pragmatic model transformations for analyzing bounded and positive responseshttps://zbmath.org/1496.621302022-11-17T18:59:28.764376Z"Tourani-Farani, Fahimeh"https://zbmath.org/authors/?q=ai:tourani-farani.fahimeh"Kazemi, Iraj"https://zbmath.org/authors/?q=ai:kazemi.irajSummary: Extensions of modeling continuously bounded and positive responses in regression contexts are often prominent. Most regression techniques incorporate a response transformation to improve underlying model fittings. A further challenge is, however, demanding to promise the transformation success. It motivated us to introduce a novel modeling strategy using the generalized Johnson system of transformations. We propose joint regression modeling of the median and precision parameters by exploiting various invertible transformations and link functions. It offers a convenient alternative to several regression models, including the normal, the popular Beta for bounded, and the log-symmetric for positive responses. Other attractive features include the iteratively reweighted-least-squares algorithm (IRLS) development to facilitate computational aspects and robust residual diagnostics to detect outlying points. Monte Carlo simulations and analysis of three real-life data sets illustrate the usefulness of our modeling strategy.Almost sure convergence of randomized urn models with application to elephant random walkhttps://zbmath.org/1496.621392022-11-17T18:59:28.764376Z"Gangopadhyay, Ujan"https://zbmath.org/authors/?q=ai:gangopadhyay.ujan"Maulik, Krishanu"https://zbmath.org/authors/?q=ai:maulik.krishanuSummary: We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and \(L^1\) convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d. replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in [\textit{K. B. Athreya} and \textit{P. E. Ney}, Branching processes. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0259.60002)] requiring \(L \log_+ L\) moments. For more general adaptive sequence of replacement matrices, a little more than \(L \log_+ L\) condition is required. Similar results based on \(L^1\) moment assumption alone has been considered independently and in parallel in [\textit{L.-X. Zhang}, ``Convergence of randomized urn models with irreducible and reducible replacement policy'', Preprint, \url{arXiv:2204.04810}]. Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in \(d\) dimension with random memory.Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observationshttps://zbmath.org/1496.621412022-11-17T18:59:28.764376Z"Dexheimer, Niklas"https://zbmath.org/authors/?q=ai:dexheimer.niklas"Strauch, Claudia"https://zbmath.org/authors/?q=ai:strauch.claudiaSummary: We consider nonparametric invariant density and drift estimation for a class of multidimensional degenerate resp. hypoelliptic diffusion processes, so-called stochastic damping Hamiltonian systems or kinetic diffusions, under anisotropic smoothness assumptions on the unknown functions. The analysis is based on continuous observations of the process, and the estimators' performance is measured in terms of the \(\sup\)-norm loss. Regarding invariant density estimation, we obtain highly nonclassical results for the rate of convergence, which reflect the inhomogeneous variance structure of the process. Concerning estimation of the drift vector, we suggest both non-adaptive and fully data-driven procedures. All of the aforementioned results strongly rely on tight uniform moment bounds for empirical processes associated to deterministic and stochastic integrals of the investigated process, which are also proven in this paper.Multidimensional parameter estimation of heavy-tailed moving averageshttps://zbmath.org/1496.621422022-11-17T18:59:28.764376Z"Ljungdahl, Mathias Mørck"https://zbmath.org/authors/?q=ai:ljungdahl.mathias-morck"Podolskij, Mark"https://zbmath.org/authors/?q=ai:podolskij.markSummary: In this article we present a parametric estimation method for certain multiparameter heavy-tailed Lévy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2,1) models, and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.Approximate maximum likelihood estimation for one-dimensional diffusions observed on a fine gridhttps://zbmath.org/1496.621432022-11-17T18:59:28.764376Z"Lu, Kevin W."https://zbmath.org/authors/?q=ai:lu.kevin-w"Paine, Phillip J."https://zbmath.org/authors/?q=ai:paine.phillip-j"Preston, Simon P."https://zbmath.org/authors/?q=ai:preston.simon-p"Wood, Andrew T. A."https://zbmath.org/authors/?q=ai:wood.andrew-t-aSummary: We consider a one-dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô-Taylor expansion of the sample path, combined with an application of the so-called \(\epsilon\)-expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high-frequency sampling framework. This method of density approximation leads to a closed-form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood-based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate methods in the literature in terms of speed, accuracy, and ease of implementation.Trajectory fitting estimation for a class of SDEs with small Lévy noiseshttps://zbmath.org/1496.621442022-11-17T18:59:28.764376Z"Zhang, Xuekang"https://zbmath.org/authors/?q=ai:zhang.xuekang"Shu, Huisheng"https://zbmath.org/authors/?q=ai:shu.huishengSummary: In this paper, we consider the problem of trajectory fitting estimation for a class of stochastic differential equations with small Lévy noises based on continuous-time observations. The consistency, the rate of convergence, and asymptotic distribution of the trajectory fitting estimator are established as a small dispersion coefficient tends to zero.Minimum distance estimation of locally stationary moving average processeshttps://zbmath.org/1496.621462022-11-17T18:59:28.764376Z"Vicuña, M. Ignacia"https://zbmath.org/authors/?q=ai:vicuna.m-ignacia"Palma, Wilfredo"https://zbmath.org/authors/?q=ai:palma.wilfredo"Olea, Ricardo"https://zbmath.org/authors/?q=ai:olea.ricardo-aSummary: The minimum distance methodology can be applied to the estimation of locally stationary moving average processes. This novel approach allows for the analysis of time series data exhibiting non-stationary behavior. The main advantages of this method are that it does not depend on the distribution of the process, can handle missing data and is computationally efficient. Some large sample properties of the new estimator are investigated, establishing its consistency and asymptotic normality. The Monte Carlo experiments presented show that the estimates behave well even for small sample sizes. The proposed methodology is illustrated by means of an application to a real-life time series of data.Asymptotic properties of mildly explosive processes with locally stationary disturbancehttps://zbmath.org/1496.621542022-11-17T18:59:28.764376Z"Hirukawa, Junichi"https://zbmath.org/authors/?q=ai:hirukawa.junichi"Lee, Sangyeol"https://zbmath.org/authors/?q=ai:lee.sangyeolSummary: In this study, we derive the limiting distribution of the least squares estimator (LSE) and the localized LSE for mildly explosive autoregressive models with locally stationary disturbance and verify that it is Cauchy as in the iid case. We also investigate the limiting distribution of two types of Dickey-Fuller unit root tests, designed for detecting a bubble period in economic time series data, and show that these tests are consistent. To evaluate the methods, we conduct a simulation study and carry out a data analysis using time series data on bitcoin prices.Vector-valued generalized Ornstein-Uhlenbeck processes: properties and parameter estimationhttps://zbmath.org/1496.621582022-11-17T18:59:28.764376Z"Voutilainen, Marko"https://zbmath.org/authors/?q=ai:voutilainen.marko"Viitasaari, Lauri"https://zbmath.org/authors/?q=ai:viitasaari.lauri"Ilmonen, Pauliina"https://zbmath.org/authors/?q=ai:ilmonen.pauliina"Torres, Soledad"https://zbmath.org/authors/?q=ai:torres.soledad"Tudor, Ciprian"https://zbmath.org/authors/?q=ai:tudor.ciprian-aSummary: Generalizations of the Ornstein-Uhlenbeck process defined through Langevin equations, such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one-dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian stationary increment noise. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter is a matrix and the noise is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the parameter matrix. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterizes essentially all multidimensional stationary processes.Maximum composite likelihood estimation for spatial extremes models of Brown-Resnick type with application to precipitation datahttps://zbmath.org/1496.621642022-11-17T18:59:28.764376Z"Kim, Moosup"https://zbmath.org/authors/?q=ai:kim.moosup"Lee, Sangyeol"https://zbmath.org/authors/?q=ai:lee.sangyeolSummary: In this study, we consider the maximum composite likelihood estimator for spatial extremes model class of Brown-Resnick type. The composite likelihood is constructed based on the weighted tail empirical process. It is shown that the proposed estimator is consistent and asymptotically normal under some regularity conditions fulfilled by the model class. We conduct Monte Carlo simulations to evaluate the estimator and apply it to the analysis of a precipitation data set.A solution to a linear integral equation with an application to statistics of infinitely divisible moving averageshttps://zbmath.org/1496.621662022-11-17T18:59:28.764376Z"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochen"Roth, Stefan"https://zbmath.org/authors/?q=ai:roth.stefan"Spodarev, Evgeny"https://zbmath.org/authors/?q=ai:spodarev.evgueniSummary: For a stationary moving average random field, a non-parametric low frequency estimator of the Lévy density of its infinitely divisible independently scattered integrator measure is given. The plug-in estimate is based on the solution \(w\) of the linear integral equation \(v(x)=\int_{\mathbb{R}^d} g(s)w(h(s)x)ds\), where \(g, h : \mathbb{R}^d \to \mathbb{R}\) are given measurable functions and \(v\) is a (weighted) \(L^2\)-function on \(\mathbb{R}\). We investigate conditions for the existence and uniqueness of this solution and give \(L^2\)-error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.Long-term prediction of the metals' prices using non-Gaussian time-inhomogeneous stochastic processhttps://zbmath.org/1496.621752022-11-17T18:59:28.764376Z"Szarek, Dawid"https://zbmath.org/authors/?q=ai:szarek.dawid"Bielak, Łukasz"https://zbmath.org/authors/?q=ai:bielak.lukasz"Wyłomańska, Agnieszka"https://zbmath.org/authors/?q=ai:wylomanska.agnieszkaSummary: Stochastic models traditionally used to describe metals' prices have proved not to be suitable to represent the dynamic behavior and time-related nature of metal markets. Rates of return are characterized by non-Gaussian and heterogeneous characteristics, which requires the use of properly adjusted models. In this paper, we introduce a stochastic model that takes under consideration the mentioned specific characteristics of the real data corresponding to the mineral commodity prices, namely the non-homogeneous character (time-dependent characteristics) and non-Gaussian distribution. The introduced model is in some sense the extension of the classical Ornstein-Uhlenbeck process (called also the Vasicek model) which was originally used to the interest rate data description. The proposed in this paper model, in contrast to the classical process, has the time-dependent parameters. This perfectly captures the time-dependent characteristics of the real data. Moreover, it is based on the general class of the skewed Student's t-distribution (SGT), which is related to the non-Gaussian behavior of the real metals' prices. This paper is a continuation of the authors' previous research where the simpler model (Chan-Karolyi-Longstaff-Sander, CKLS) based on the SGT distribution was proposed. We demonstrate here the step-by-step procedure of the time-dependent parameters' estimation and check its effectiveness by using the simulated data. Finally, based on the real-time series analysis, we demonstrate that the proposed stochastic model is universal and can be applied to metals' prices description for the long-term prediction.Semiparametric Bayesian forecasting of spatiotemporal earthquake occurrenceshttps://zbmath.org/1496.621982022-11-17T18:59:28.764376Z"Ross, Gordon J."https://zbmath.org/authors/?q=ai:ross.gordon-j"Kolev, Aleksandar A."https://zbmath.org/authors/?q=ai:kolev.aleksandar-aSummary: The Epidemic Type Aftershock Sequence (ETAS) model is a self-exciting point process which is used to model and forecast the occurrence of earthquakes in a geographical region. The ETAS model assumes that the occurrence of mainshock earthquakes follows an inhomogeneous spatial point process, with their aftershock earthquakes modelled via a separate triggering kernel. Most previous studies of the ETAS model have relied on point estimates of the model parameters, due to the complexity of the likelihood function and the difficulty in estimating an appropriate spatial mainshock distribution. In order to take estimation uncertainty into account, we instead propose a fully Bayesian formulation of the ETAS model, which uses a nonparametric Dirichlet process mixture prior to capture the spatial mainshock process, and show how efficient parameter inference can be carried out using auxiliary latent variables. We demonstrate how our model can be used for medium-term earthquake forecasts in a number of geographical regions.Parameter estimation for a discretely observed population process under Markov-modulationhttps://zbmath.org/1496.622162022-11-17T18:59:28.764376Z"de Gunst, Mathisca"https://zbmath.org/authors/?q=ai:de-gunst.mathisca-c-m"Knapik, Bartek"https://zbmath.org/authors/?q=ai:knapik.bartek"Mandjes, Michel"https://zbmath.org/authors/?q=ai:mandjes.michel"Sollie, Birgit"https://zbmath.org/authors/?q=ai:sollie.birgitSummary: A Markov-modulated independent sojourn process is a population process in which individuals arrive according to a Poisson process with Markov-modulated arrival rate, and leave the system after an exponentially distributed time. A procedure is developed to estimate the parameters of such a system, including those related to the modulation. It is assumed that the number of individuals in the system is observed at equidistant time points only, whereas the modulating Markov chain cannot be observed at all. An algorithm is set up for finding maximum likelihood estimates, based on the EM algorithm and containing a forward -- backward procedure for computing the conditional expectations. To illustrate the performance of the algorithm the results of an extensive simulation study are presented.Stochastic partial differential equations for computer vision with uncertain datahttps://zbmath.org/1496.650032022-11-17T18:59:28.764376Z"Preusser, Tobias"https://zbmath.org/authors/?q=ai:preusser.tobias"Kirby, Robert M."https://zbmath.org/authors/?q=ai:kirby.robert-m-ii"Pätz, Torben"https://zbmath.org/authors/?q=ai:patz.torbenPublisher's description: In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be -- and more and more frequently are -- taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate.
The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness.
This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.Exponential mean-square stability properties of stochastic linear multistep methodshttps://zbmath.org/1496.650072022-11-17T18:59:28.764376Z"Buckwar, Evelyn"https://zbmath.org/authors/?q=ai:buckwar.evelyn"D'Ambrosio, Raffaele"https://zbmath.org/authors/?q=ai:dambrosio.raffaeleThe paper studies multistep methods for nonlinear stochastic differential equations. The definition of exponential mean-square contractivity is given for any two solutions of SDE. The authors develop a general analysis of the exponential mean-square stability properties of the general family of stochastic two-step methods. The following methods are analyzed: Adams-Moulton, Milne-Simpson, BDF2.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Error estimate for the approximate solution to multivariate feedback particle filterhttps://zbmath.org/1496.650082022-11-17T18:59:28.764376Z"Dong, Wenhui"https://zbmath.org/authors/?q=ai:dong.wenhui"Gao, Xingbao"https://zbmath.org/authors/?q=ai:gao.xingbaoSummary: In this paper, based on the assumption that the gain function \(K\) has been optimally obtained in the multivariate feedback particle filter (FPF), we focus on the error estimate for the approximate solutions to the particle's density evolution equation, which is actually the forward Kolmogorov equation (FKE) satisfied by the ``particle population''. The approximation is essentially the unnormalized density of the states conditioning on the discrete observations with the given time discretization. Mainly owing to the representation of Brownian bridges for the Brownian motion, and the assumption on the coercivity condition, we prove that the mean square error of the approximate solution is of order equal to the square root of the time interval.Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equationshttps://zbmath.org/1496.650092022-11-17T18:59:28.764376Z"Eisenmann, Monika"https://zbmath.org/authors/?q=ai:eisenmann.monika"Kovács, Mihály"https://zbmath.org/authors/?q=ai:kovacs.mihaly"Kruse, Raphael"https://zbmath.org/authors/?q=ai:kruse.raphael"Larsson, Stig"https://zbmath.org/authors/?q=ai:larsson.stigSummary: In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [\textit{R. H. Nochetto} et al., Commun. Pure Appl. Math. 53, No. 5, 525--589 (2000; Zbl 1021.65047)]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic \(p\)-Laplace equation.Weak convergence of the L1 scheme for a stochastic subdiffusion problem driven by fractionally integrated additive noisehttps://zbmath.org/1496.650102022-11-17T18:59:28.764376Z"Hu, Ye"https://zbmath.org/authors/?q=ai:hu.ye"Li, Changpin"https://zbmath.org/authors/?q=ai:li.changpin"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubinSummary: The weak convergence of a fully discrete scheme for approximating a stochastic subdiffusion problem driven by fractionally integrated additive noise is studied. The Caputo fractional derivative is approximated by the L1 scheme and the Riemann-Liouville fractional integral is approximated with the first order convolution quadrature formula. The noise is discretized by using the Euler method and the spatial derivative is approximated with the linear finite element method. Based on the nonsmooth data error estimates of the corresponding deterministic problem, the weak convergence orders of the fully discrete schemes for approximating the stochastic subdiffusion problem driven by fractionally integrated additive noise are proved by using the Kolmogorov equation approach. Numerical experiments are given to show that the numerical results are consistent with the theoretical results.Weak convergence of Euler scheme for SDEs with low regular drifthttps://zbmath.org/1496.650112022-11-17T18:59:28.764376Z"Suo, Yongqiang"https://zbmath.org/authors/?q=ai:suo.yongqiang"Yuan, Chenggui"https://zbmath.org/authors/?q=ai:yuan.chenggui"Zhang, Shao-Qin"https://zbmath.org/authors/?q=ai:zhang.shaoqinSummary: In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with low regular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in some fractional Sobolev space.Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noisehttps://zbmath.org/1496.650122022-11-17T18:59:28.764376Z"Yang, Zhiwei"https://zbmath.org/authors/?q=ai:yang.zhiwei"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Zhang, Zhongqiang"https://zbmath.org/authors/?q=ai:zhang.zhongqiang"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We prove the existence and uniqueness of the solution to a variable-order fractional stochastic differential equation driven by a multiplicative white noise, which describes the random phenomena with nonlocal effect. We further develop a Euler-Maruyama scheme and prove the strong convergence of the scheme. Numerical experiments are presented to substantiate the mathematical analysis.Temporal second-order difference methods for solving multi-term time fractional mixed diffusion and wave equationshttps://zbmath.org/1496.651112022-11-17T18:59:28.764376Z"Du, Rui-lian"https://zbmath.org/authors/?q=ai:du.ruilian"Sun, Zhi-zhong"https://zbmath.org/authors/?q=ai:sun.zhizhongSummary: This article deals with an establishment and sharp theoretical analysis of a numerical scheme devised for solving the multi-dimensional multi-term time fractional mixed diffusion and wave equations. The governing equation contains both fractional diffusion term and fractional wave term which make the numerical analysis challenging. With the help of the method of order reduction, we convert the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively, and then develop \(L2\)-\(1_\sigma\) formula for solving the latter problem. In addition, the formula is used to numerically solve the time distributed-order diffusion and wave equations. The stability and convergence of these numerical schemes are rigorously analyzed by the energy method. The convergence rates are of order two in both time and space. A difference scheme on nonuniform time grids is also constructed for solving the problem with weak regularity at the initial time. Finally, we illustrate our results with some examples.Strong \(L^2\) convergence of time numerical schemes for the stochastic two-dimensional Navier-Stokes equationshttps://zbmath.org/1496.651402022-11-17T18:59:28.764376Z"Bessaih, Hakima"https://zbmath.org/authors/?q=ai:bessaih.hakima"Millet, Annie"https://zbmath.org/authors/?q=ai:millet.annieSummary: We prove that some time discretization schemes for the two-dimensional Navier-Stokes equations on the torus subject to a random perturbation converge in \(L^2(\varOmega )\). This refines previous results that established the convergence only in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme we can prove strong convergence of fully implicit and semiimplicit temporal Euler discretizations and of a splitting scheme. The speed of the \(L^2(\varOmega )\) convergence depends on the diffusion coefficient and on the viscosity parameter.Space-time approximation of stochastic \(p\)-Laplace-type systemshttps://zbmath.org/1496.651562022-11-17T18:59:28.764376Z"Breit, Dominic"https://zbmath.org/authors/?q=ai:breit.dominic"Hofmanová, Martina"https://zbmath.org/authors/?q=ai:hofmanova.martina"Loisel, Sébastien"https://zbmath.org/authors/?q=ai:loisel.sebastienAn efficient hybrid method for uncertainty quantificationhttps://zbmath.org/1496.651712022-11-17T18:59:28.764376Z"Wahlsten, Markus"https://zbmath.org/authors/?q=ai:wahlsten.markus"Ålund, Oskar"https://zbmath.org/authors/?q=ai:alund.oskar"Nordström, Jan"https://zbmath.org/authors/?q=ai:nordstrom.janThe paper focuses on coupling intrusive and non-intrusive uncertainty quantification methods for hyperbolic systems of equations, which are applied in different parts of the computational domain separated by interfaces. The intrusive method is based on the stochastic Galerkin projection combined with a polynomial chaos basis, which is efficient for slowly changing stochastic problems. The non-intrusive method employs a quadrature rule and the probability density functions of the uncertainties, which is well-suited for rapidly varying problems. The proposed coupling, based on a suitable projection onto an intermediate glue grid and penalization, is shown to be stable and accurate. Numerical experiments are presented for one- and two-dimensional domains to illustrate the efficiency and applicability of the method.
Reviewer: Dana Černá (Liberec)Sharp error estimates on a stochastic structure-preserving scheme in computing effective diffusivity of 3D chaotic flowshttps://zbmath.org/1496.651922022-11-17T18:59:28.764376Z"Wang, Zhongjian"https://zbmath.org/authors/?q=ai:wang.zhongjian"Xin, Jack"https://zbmath.org/authors/?q=ai:xin.jack-x"Zhang, Zhiwen"https://zbmath.org/authors/?q=ai:zhang.zhiwenOptimal strong convergence of finite element methods for one-dimensional stochastic elliptic equations with fractional noisehttps://zbmath.org/1496.652082022-11-17T18:59:28.764376Z"Cao, Wanrong"https://zbmath.org/authors/?q=ai:cao.wanrong"Hao, Zhaopeng"https://zbmath.org/authors/?q=ai:hao.zhaopeng"Zhang, Zhongqiang"https://zbmath.org/authors/?q=ai:zhang.zhongqiangSummary: We investigate the strong convergence order of piecewise linear finite element methods for a class of one-dimensional semilinear stochastic elliptic equations with additive fractional white noise. For the Hurst index \(H\in (0,1)\), we approximate the fractional Brownian motion by two spectral expansions. We show that the resulting schemes are of order \(H+1\) in the mean-square sense if the element size \(h\) is taken proportionally to the truncation parameters in the spectral approximations. Numerical results confirm our theoretical prediction.Asymptotic connectedness of random interval graphs in a one dimensional data delivery problemhttps://zbmath.org/1496.682612022-11-17T18:59:28.764376Z"Andrade Sernas, Caleb Erubiel"https://zbmath.org/authors/?q=ai:andrade-sernas.caleb-erubiel"Calvillo Vives, Gilberto"https://zbmath.org/authors/?q=ai:calvillo-vives.gilberto"Manrique Mirón, Paulo Cesar"https://zbmath.org/authors/?q=ai:manrique-miron.paulo-cesar"Treviño Aguilar, Erick"https://zbmath.org/authors/?q=ai:aguilar.erick-trevinoSummary: In this work we present a probabilistic analysis of random interval graphs associated with randomly generated instances of the Data Delivery on a Line Problem (DDLP)
[\textit{J. Chalopin} et al., Lect. Notes Comput. Sci. 8573, 423--434 (2014; Zbl 1411.68157)].
Random Interval Graphs have been previously studied by
\textit{E. R. Scheinerman} [Discrete Math. 82, No. 3, 287--302 (1990; Zbl 0699.05051)].
However, his model and ours provide different ways to generate the graphs. Our model is defined by how the agents in the DDLP may move, thus its importance goes beyond the intrinsic interest of random graphs and has to do with the complexity of a combinatorial optimization problem which has been proven to be NP-complete
[Zbl 1411.68157].
We study the relationship between solvability of a random instance of the DDLP with respect to its associated interval graph connectedness. This relationship is important because through probabilistic analysis we prove that despite the NP-completeness of DDLP, there are classes of instances that can be solved polynomially.
For the entire collection see [Zbl 1478.60006].Lagrangian chaos and scalar advection in stochastic fluid mechanicshttps://zbmath.org/1496.760402022-11-17T18:59:28.764376Z"Bedrossian, Jacob"https://zbmath.org/authors/?q=ai:bedrossian.jacob"Blumenthal, Alex"https://zbmath.org/authors/?q=ai:blumenthal.alex"Punshon-Smith, Sam"https://zbmath.org/authors/?q=ai:punshon-smith.samSummary: We study the Lagrangian flow associated to velocity fields arising from various models of stochastic fluid mechanics. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive (\textit{almost surely}, \textit{all} particle trajectories are simultaneously exponentially sensitive with respect to initial conditions). Our main results are for the Navier-Stokes equations on \(\mathbb{T}^2\) and the hyper-viscous regularized Navier-Stokes equations on \(\mathbb{T}^3\) (at arbitrary fixed Reynolds number and hyper-viscosity parameters), subject to white-in-time, \(H^s\)-in-space stochastic forcing which is nondegenerate at high frequencies. Using these results, we further make a mathematically rigorous study of ``passive scalar turbulence''. To this end, we study statistically stationary solutions to the passive scalar advection-diffusion advected by one of these velocity fields and subjected to a white-in-time random source. We show that the chaotic behavior of Lagrangian dynamics implies a type of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies Yaglom's law of scalar turbulence -- the universal scaling law analogous to the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems in infinite dimensions, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.Transfer of orbital angular momentum states of light in \(\Lambda\)-type quantum systemhttps://zbmath.org/1496.810302022-11-17T18:59:28.764376Z"Ye, Fuqiu"https://zbmath.org/authors/?q=ai:ye.fuqiuSummary: We observe the quantum interference impact resulted from incoherent pumping fields on orbital angular momentum (OAM) switch among optical fields in three-level quantum systems. We recall a \(\Lambda\)-kind three level quantum system in which the system is to start with prepared in top excited state. We discover that for suitable parameters of incoherent pumping rates and quantum interference term, the exchange of the optical vortices is possible. We understand that the exchange performance is excessive sufficient whilst we take into account the intense incoherent pumping fields and quantum interference term, respectively. Also, we observe the spatial established of the absorption of the signal light for identifying the OAM number of the vortex probe light. We understand that the absorption of the non-vortex signal light relies upon to the azimuthal phase of the probe vortex light because of presence of quantum interference from incoherent fields.Quantum steering and quantum discord under noisy channels and entanglement swappinghttps://zbmath.org/1496.810362022-11-17T18:59:28.764376Z"Rosario, Pedro"https://zbmath.org/authors/?q=ai:rosario.pedro"Ducuara, Andrés F."https://zbmath.org/authors/?q=ai:ducuara.andres-f"Susa, Cristian E."https://zbmath.org/authors/?q=ai:susa.cristian-eSummary: Quantum entanglement, quantum discord, and EPR-steering are properties which are considered as valuable resources for fuelling quantum information-theoretic protocols. EPR-steering is a correlation weaker than nonlocality (in the Bell's sense) and yet stronger than entanglement. Quantum discord on the other hand, captures non-classical behaviour beyond that of entanglement, and its study has remained of active research interest during the past two decades. Exploring the behaviour of these quantum correlations in different physical scenarios, like those simulated by open quantum systems, is therefore of crucial importance for understanding their viability for quantum technologies. In this work, we analyse the behaviour of EPR-steering, entanglement, and quantum discord, for partially entangled two-qubit states with coloured noise, introduced by \textit{M. L. Ameida} et al. [``Noise robustness of the nonlocality of entangled quantum states'', Phys. Rev. Lett. 99, No. 4, Article ID 040403, 4 p. (2007; \url{doi:10.1103/PhysRevLett.99.040403})], under various quantum processes. First, we consider the three \textit{noisy channel} scenarios of; phase damping, generalised amplitude damping and stochastic dephasing channel. Second, we explore their behaviour in an \textit{entanglement swapping} scenario. We quantify EPR-steering by means of an inequality with three-input two-output measurement settings, and address quantum discord as the interferometric power of quantum states. We discuss the sudden death of steering and entanglement induced by the noisy processes. Additionally, in the case of generalised amplitude damping, a death and revival behaviour can be interpreted in terms of one of the noise's parameters. Second, we contrast the fact that noisy channels in general reduce the amount of correlations present in the system, with the swapping protocol, which displays scenarios where these quantum correlations can be enhanced with respect to the correlations at the initial stage of the protocol. In particular, we present a trade-off between the post-swap amount of correlations and their probability of occurrence.Effect of noise in the quantum bidirectional direct communication protocol using non-maximally entangled stateshttps://zbmath.org/1496.810392022-11-17T18:59:28.764376Z"Ramachandran, Meera"https://zbmath.org/authors/?q=ai:ramachandran.meera"Balakrishnan, S."https://zbmath.org/authors/?q=ai:balakrishnan.sivasubramanya|balakrishnan.suhrid|balakrishnan.sivaraman|balakrishnan.srinivasan|balakrishnan.sarasija-perurkada|balakrishnan.srivatsan|balakrishnan.swaminathan|balakrishnan.sreeram|balakrishnan.s-n|balakrishnan.sudha|balakrishnan.shankar|balakrishnan.subramaniam|balakrishnan.shanmugamSummary: Possibility of using non-maximally entangled states in quantum bidirectional direct communication has been shown recently by \textit{A. Srikanth} and \textit{S. Balakrishnan} [``Controller-independent quantum bidirectional communication using non-maximally entangled states'', Quantum Inf. Process. 19, Paper No. 133, 11 p. (2020; \url{doi:10.1007/s11128-020-02628-2})]. The effect of noise in this protocol is analyzed by considering amplitude and phase damping models. Suitable combinations of messages and initial states are identified to minimize the effect of noise in the protocol. Further, we have shown the possibility of demarking the effects due to noise and the intruder.Dynamics of quantum speed limit time for correlated and uncorrelated noise channelshttps://zbmath.org/1496.810412022-11-17T18:59:28.764376Z"Awasthi, Natasha"https://zbmath.org/authors/?q=ai:awasthi.natasha"Joshi, Dheeraj Kumar"https://zbmath.org/authors/?q=ai:joshi.dheeraj-kumar"Sachdev, Surbhi"https://zbmath.org/authors/?q=ai:sachdev.surbhiSummary: We investigate the dynamics of quantum speed limit time under the effect of Markovian and non-Markovian dynamics. Memory effect plays a fundamental role in the dynamics of open quantum system. In the first case quantum channel has memory there exist correlations between sucessive uses of channels. Such type of channel called correalted channel. We study quantum speed limit time for Markovian and non-Markovian dynamics. We discuss measure of QSL, how correlations affect the rate of quantum speed limit. In this work find one connection between non-Markovianty and quantum speed limit. We show that the non Markovian evolution can speed up quantum evolution, therefore lead to smaller quantum speed limit time. We establish connection between information loss, quantum speed limit time that might be helpful for further investigation in quantum processing theoretical task.Speed of quantum evolution for correlated quantum noisehttps://zbmath.org/1496.810422022-11-17T18:59:28.764376Z"Haseli, Soroush"https://zbmath.org/authors/?q=ai:haseli.soroush"Hadipour, Maryam"https://zbmath.org/authors/?q=ai:hadipour.maryamSummary: Memory effects play an important role in the theory of open quantum systems. There are two completely independent insights about memory for quantum channels. In quantum information theory, the memory of the quantum channel is depicted by the correlations between consecutive uses of the channel on a set of quantum systems. In the theory of open quantum systems memory effects result from correlations which are created during the quantum evolution. Here, we study the behavior of the actual speed of the quantum evolution under the effects of the correlated channel. The speed of quantum evolution indicates how a quantum system can be rapidly transformed from an initial state to a secondary state in a quantum evolution. In this work, we consider two correlated channels: Correlated phase damping channel and correlated dephasing channel with colored noise. We will show that the speed of quantum evolution for correlated channel is lower than the speed in uncorrelated channel. We will also show that the speed of quantum evolution in non-Markovian processes and in the presence of dynamical memory effects is greater than speed in the Markovian process and in the absence of dynamical memory effects.A quantum-mechanical derivation of the multivariate central limit theorem for Markov chainshttps://zbmath.org/1496.810582022-11-17T18:59:28.764376Z"Ricci, Leonardo"https://zbmath.org/authors/?q=ai:ricci.leonardoSummary: The central limit theorem for Markov chains is widely used, in particular in its pristine univariate form. As far as the multivariate case is concerned, a few proofs exist, which depend on different assumptions and require advanced mathematical and statistical tools. Here a novel proof is presented that, starting from the standard condition of regularity only, relies on time-independent quantum-mechanical perturbation theory. The result, which is obtained by using techniques that are typical of physics, is expected to enhance the usability of this cornerstone theorem, especially in nonlinear dynamics and physics of complex systems.On a generalized central limit theorem and large deviations for homogeneous open quantum walkshttps://zbmath.org/1496.810602022-11-17T18:59:28.764376Z"Carbone, Raffaella"https://zbmath.org/authors/?q=ai:carbone.raffaella"Girotti, Federico"https://zbmath.org/authors/?q=ai:girotti.federico"Hernandez, Anderson Melchor"https://zbmath.org/authors/?q=ai:hernandez.anderson-melchorSummary: We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.Factoring discrete-time quantum walks on distance regular graphs into continuous-time quantum walkshttps://zbmath.org/1496.810612022-11-17T18:59:28.764376Z"Zhan, Hanmeng"https://zbmath.org/authors/?q=ai:zhan.hanmengSummary: We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph \(X\). Given that \(X\) has diameter \(d\) and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on \(X\) is a product of at most \(d\) commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of \(X\). We also obtain a similar factorization for any graph \(X\) in a Bose Mesner algebra.Wigner function as a detector of entanglement in open two coupled Inas semiconductor quantum dotshttps://zbmath.org/1496.810632022-11-17T18:59:28.764376Z"Mansour, H. Ait"https://zbmath.org/authors/?q=ai:mansour.hicham-ait"Siyouri, F-Z."https://zbmath.org/authors/?q=ai:siyouri.f-zSummary: We tested the ability of Wigner function to reveal and capture the quantum entanglement presents in two coupled semiconductor InAs quantum dots that independently interact with dephasing reservoirs. In this respect, we analyze their evolution against the temperature parameter as well as against the dimensionless time in both Markovian and non-Markovian environments. Further, we compare their amounts and their behaviors under the Förster interaction effect. In particular, we show that for large values of dimensionless time and at higher temperature, unlike the full disappear of entanglement the positive part of Wigner function still survives. Moreover, we show that the Wigner function volume is influenced by the variation of the Förster interaction, the temperature and the non-Markovianity degree. Nevertheless, its ability to reveal the quantum entanglement presents inside two coupled semiconductor quantum dots is still kept.Optical effects of domain wallshttps://zbmath.org/1496.810682022-11-17T18:59:28.764376Z"Khoze, Valentin V."https://zbmath.org/authors/?q=ai:khoze.valentin-v"Milne, Daniel L."https://zbmath.org/authors/?q=ai:milne.daniel-lSummary: Domain walls arise in theories where there is spontaneous symmetry breaking of a discrete symmetry such as \(\mathbb{Z}_N\) and are a feature of many BSM models. In this work we consider the possibility of detecting domain walls through their optical effects and specify three different methods of coupling domain walls to the photon. We consider the effects of these couplings in the context of gravitational wave detectors, such as LIGO, and examine the sensitivity of these experiments to domain wall effects. In cases where gravitational wave detectors are not sensitive we examine our results in the context of axion experiments and show how effects of passing domain walls can be detected at interferometers searching for an axion signal.Basic properties of a mean field laser equationhttps://zbmath.org/1496.810732022-11-17T18:59:28.764376Z"Fagnola, Franco"https://zbmath.org/authors/?q=ai:fagnola.franco"Mora, Carlos M."https://zbmath.org/authors/?q=ai:mora.carlos-mMoment dynamics and observer design for a class of quasilinear quantum stochastic systemshttps://zbmath.org/1496.810752022-11-17T18:59:28.764376Z"Vladimirov, Igor G."https://zbmath.org/authors/?q=ai:vladimirov.igor-g"Petersen, Ian R."https://zbmath.org/authors/?q=ai:petersen.ian-rCorrelation decay for hard spheres via Markov chainshttps://zbmath.org/1496.820052022-11-17T18:59:28.764376Z"Helmuth, Tyler"https://zbmath.org/authors/?q=ai:helmuth.tyler"Perkins, Will"https://zbmath.org/authors/?q=ai:perkins.will"Petti, Samantha"https://zbmath.org/authors/?q=ai:petti.samanthaSummary: We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions three and higher. As the dimension tends to infinity, our improvements are by factors of 2 and 1.7, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.Percolation of wormshttps://zbmath.org/1496.820102022-11-17T18:59:28.764376Z"Ráth, Balázs"https://zbmath.org/authors/?q=ai:rath.balazs"Rokob, Sándor"https://zbmath.org/authors/?q=ai:rokob.sandorSummary: We introduce a new correlated percolation model on the \(d\)-dimensional lattice \(\mathbb{Z}^d\) called the \textit{random length worms model}. Assume given a probability distribution on the set of positive integers (the length distribution) and \(v \in (0, \infty)\) (the intensity parameter). From each site of \(\mathbb{Z}^d\) we start \(\operatorname{POI} (v)\) independent simple random walks with this length distribution. We investigate the connectivity properties of the set \(\mathcal{S}^v\) of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then \(\mathcal{S}^v\) undergoes a percolation phase transition as \(v\) varies. Our main contribution is a sufficient condition on the length distribution which guarantees that \(\mathcal{S}^v\) percolates for all \(v > 0\) if \(d \geq 5\). E.g., if the probability mass function of the length distribution is
\[
m (\ell) = c \cdot \ln (\ln (\ell))^\varepsilon / (\ell^3 \ln (\ell)) \mathbb{1} [\ell \geq \ell_0]
\] for some \(\ell_0 > e^e\) and \(\varepsilon > 0\) then \(\mathcal{S}^v\) percolates for all \(v > 0\). Note that the second moment of this length distribution is only ``barely'' infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the \textit{Poisson zoo} and argue that the percolative behaviour of the random length worms model is quite close to being ``extremal'' in this family of models.Physical mechanism of equiprobable exclusion network with heterogeneous interactions in phase transitions: analytical analyses of steady state evolving from initial statehttps://zbmath.org/1496.820182022-11-17T18:59:28.764376Z"Wang, Yu-Qing"https://zbmath.org/authors/?q=ai:wang.yuqing"Wang, Chao-Fan"https://zbmath.org/authors/?q=ai:wang.chao-fan"Wang, Hao-Tian"https://zbmath.org/authors/?q=ai:wang.haotian"Du, Min-Xuan"https://zbmath.org/authors/?q=ai:du.min-xuan"Wang, Bing-Hong"https://zbmath.org/authors/?q=ai:wang.binghongSummary: Being a vital two-dimensional multibody interacting particle system in nonlinear science and complex systems, exclusion network fuses totally asymmetric simple exclusion process into underlying complex network dynamics. This study constructs equiprobable exclusion network with heterogeneous interactions by introducing randomly generated interaction rates on each random path. Nodes are equivalent to subnetworks modelled by periodic TASEPs. Analytical solutions of typical order parameters are obtained by exploring dynamical transitions among configuration probabilities validated by meticulous balance theory. Physical mechanisms of underlying exclusion network dynamics are revealed by discussing TASEP with boundaries and Langmuir kinetics. New analytical method named as isoline analyses on mechanisms of spatial correlation and spatiotemporal evolution is proposed. Phase boundaries between initial state and steady state are analytically solved, which have a high agreement with simulations. Fruitful mechanisms of system transiting from initial phase to steady phases are discovered. It will have theoretical and practical value of deeply understanding evolution laws of cluster dynamics of self-driven particles and exploring non-equilibrium phase transitions in active systems.Non-isothermal squeeze film damping in the test of gravitational inverse-square lawhttps://zbmath.org/1496.830122022-11-17T18:59:28.764376Z"Ke, Jun"https://zbmath.org/authors/?q=ai:ke.jun"Luo, Jie"https://zbmath.org/authors/?q=ai:luo.jie"Tan, Yu-Jie"https://zbmath.org/authors/?q=ai:tan.yujie"Liu, Zhe"https://zbmath.org/authors/?q=ai:liu.zhe"Shao, Cheng-Gang"https://zbmath.org/authors/?q=ai:shao.chenggang"Yang, Shan-Qing"https://zbmath.org/authors/?q=ai:yang.shan-qingGeneralized geodesic deviation in de Sitter spacetimehttps://zbmath.org/1496.830142022-11-17T18:59:28.764376Z"Waldstein, Isaac Raj"https://zbmath.org/authors/?q=ai:waldstein.isaac-raj"Brown, J. David"https://zbmath.org/authors/?q=ai:brown.j-davidUniformly accelerated Brownian oscillator in (2+1)D: temperature-dependent dissipation and frequency shifthttps://zbmath.org/1496.830352022-11-17T18:59:28.764376Z"Moustos, Dimitris"https://zbmath.org/authors/?q=ai:moustos.dimitrisSummary: We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski spacetime. We treat the detector as an open quantum system and employ a quantum Langevin equation to describe its time evolution, with the field, which is characterized by a frequency-independent spectral density, acting as a stochastic force. We investigate a point-like detector moving with constant acceleration through the Minkowski vacuum and an inertial one immersed in a thermal reservoir at the Unruh temperature, exploring the implications of the well-known non-equivalence between the two cases on their dynamics. We find that both the accelerated detector's dissipation rate and the shift of its frequency caused by the coupling to the field bath depend on the acceleration temperature. Interestingly enough this is not only in contrast to the case of inertial motion in a heat bath but also to any analogous quantum Brownian motion model in open systems, where dissipation and frequency shifts are not known to exhibit temperature dependencies. Nonetheless, we show that the fluctuating-dissipation theorem still holds for the detector-field system and in the weak-coupling limit an accelerated detector is driven at late times to a thermal equilibrium state at the Unruh temperature.Near equilibrium fluctuations for supermarket models with growing choiceshttps://zbmath.org/1496.900222022-11-17T18:59:28.764376Z"Bhamidi, Shankar"https://zbmath.org/authors/?q=ai:bhamidi.shankar"Budhiraja, Amarjit"https://zbmath.org/authors/?q=ai:budhiraja.amarjit-s"Dewaskar, Miheer"https://zbmath.org/authors/?q=ai:dewaskar.miheerSummary: We consider the supermarket model in the usual Markovian setting where jobs arrive at rate \(n\lambda_n\) for some \(\lambda_n > 0\), with \(n\) parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among \(d_n \leq n\) randomly selected service queues. We show that when \(d_n\to \infty\) and \(\lambda_n\to \lambda \in (0,\infty)\), under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by \(\lambda\). Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when \(\lambda_n\to 1\). Previous papers, for example, [\textit{D. Mukherjee} et al., Stoch. Syst. 8, No. 4, 265--292 (2018; Zbl 1446.60073)] have considered the regime \(\frac{d_n}{\sqrt{n}\log n}\to \infty\) while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of \(d_n\). In particular, we consider the three canonical regimes (a) \(d_n /\sqrt{n}\to 0\); (b) \(d_n /\sqrt{n}\to c\in (0,\infty)\) and, (c) \(d_n /\sqrt{n}\to \infty\). In all three regimes, we show, by establishing suitable functional limit theorems, that (under conditions on \(\lambda_n)\) fluctuations of the state process about its near equilibrium are of order \(n^{-1/2}\) and are governed asymptotically by a one-dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case, we get a linear diffusion; in the second case, we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case \(d_n /(\sqrt{n}\log n)\to \infty\), our work gives alternative proofs for the universality results established in [loc. cit.].Analysis of a single server queue in a multi-phase random environment with working vacations and customers' impatiencehttps://zbmath.org/1496.900232022-11-17T18:59:28.764376Z"Bouchentouf, Amina Angelika"https://zbmath.org/authors/?q=ai:bouchentouf.amina-angelika"Guendouzi, Abdelhak"https://zbmath.org/authors/?q=ai:guendouzi.abdelhak"Meriem, Houalef"https://zbmath.org/authors/?q=ai:meriem.houalef"Shakir, Majid"https://zbmath.org/authors/?q=ai:shakir.majidSummary: In this paper, we analyze an \(M/M/1\) queueing system under both single and multiple working vacation policies, multiphase random environment, waiting server, balking and reneging. When the system is in operative phase \(j = 1,2,\ldots,K\), customers are served one by one. Whenever the system becomes empty, the server waits a random amount of time before taking a vacation, causing the system to move to working vacation phase 0 at which new arrivals are served at a lower rate. Using the probability generating function method, we obtain the distribution for the steady-state probabilities of the system. Then, we derive important performance measures of the queueing system. Finally, some numerical examples are illustrated to show the impact of system parameters on performance measures of the queueing system.A discussion on the optimality of bulk entry queue with differentiated hiatuseshttps://zbmath.org/1496.900242022-11-17T18:59:28.764376Z"Vadivukarasi, Manickam"https://zbmath.org/authors/?q=ai:vadivukarasi.manickam"Kalidass, Kaliappan"https://zbmath.org/authors/?q=ai:kalidass.kaliappanSummary: We consider Markovian differentiated hiatuses queues with bulk entries. With the help of the matrix geometric method, we discuss the stability condition for the existence of the steady-state solution of our model and we obtain the stationary system size by using a probability generating function. The stochastic decomposition form of stationary system size and the waiting time distribution of an arbitrary beneficiary are also analysed. Furthermore, we perform the expense analysis using the particle swarm optimization technique and we obtain the optimality of service rate and hiatus rate. Finally, we study the effects of changes in the parameters on some important performance measures of the system through numerical observations.Stochastic interest rate modeling with fixed income derivative pricinghttps://zbmath.org/1496.910012022-11-17T18:59:28.764376Z"Privault, Nicolas"https://zbmath.org/authors/?q=ai:privault.nicolasPublisher's description: This book introduces the mathematics of stochastic interest rate modeling and the pricing of related derivatives, based on a step-by-step presentation of concepts with a focus on explicit calculations. The types of interest rates considered range from short rates to forward rates such as LIBOR and swap rates, which are presented in the HJM and BGM frameworks. The pricing and hedging of interest rate and fixed income derivatives such as bond options, caps, and swaptions, are treated using forward measure techniques. An introduction to default bond pricing and an outlook on model calibration are also included as additional topics.
This third edition represents a significant update on the second edition published by World Scientific in 2012. Most chapters have been reorganized and largely rewritten with additional details and supplementary solved exercises. New graphs and simulations based on market data have been included, together with the corresponding R codes.
This new edition also contains 75 exercises and 4 problems with detailed solutions, making it suitable for advanced undergraduate and graduate level students.
See the reviews of the first and second editions in [Zbl 1213.91003; Zbl 1248.91002].Convergence of deep fictitious play for stochastic differential gameshttps://zbmath.org/1496.910152022-11-17T18:59:28.764376Z"Han, Jiequn"https://zbmath.org/authors/?q=ai:han.jiequn"Hu, Ruimeng"https://zbmath.org/authors/?q=ai:hu.ruimeng"Long, Jihao"https://zbmath.org/authors/?q=ai:long.jihaoSummary: Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large \(N\)-player asymmetric stochastic differential games [\textit{J. Han} and \textit{R. Hu}, ``Deep fictitious play for finding Markovian Nash equilibrium in multi-agent games'', Proc. Mach. Learn. Res. (PMLR) 107, 221--245 (2020)]. By incorporating the idea of fictitious play, the algorithm decouples the game into \(N\) sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an \(\epsilon\)-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.The card guessing game: a generating function approachhttps://zbmath.org/1496.910362022-11-17T18:59:28.764376Z"Krityakierne, Tipaluck"https://zbmath.org/authors/?q=ai:krityakierne.tipaluck"Thanatipanonda, Thotsaporn Aek"https://zbmath.org/authors/?q=ai:thanatipanonda.thotsaporn-aekSummary: Consider a card guessing game with complete feedback in which a deck of \(n\) cards ordered \(1, \dots, n\) is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at a time under the optimal strategy until no cards remain. We provide an expression for the expected number of correct guesses with arbitrary number of terms, an accuracy improvement over the results of \textit{P. Liu} [Discrete Appl. Math. 288, 270--278 (2021; Zbl 1448.91067)].
In addition, using generating functions, we give a unified framework for systematically calculating higher-order moments. Although the extension of the framework to \(k \geq 2\) shuffles is not immediately straightforward, we are able to settle a long-standing McGrath's conjectured optimal strategy described in [\textit{D. Bayer} and \textit{P. Diaconis}, Ann. Appl. Probab. 2, No. 2, 294--313 (1992; Zbl 0757.60003)] by showing that the optimal guessing strategy for \(k = 1\) riffle shuffle does not necessarily apply to \(k \geq 2\) shuffles.Pandemic-type failures in multivariate Brownian risk modelshttps://zbmath.org/1496.910392022-11-17T18:59:28.764376Z"Dȩbicki, Krzysztof"https://zbmath.org/authors/?q=ai:debicki.krzysztof"Hashorva, Enkelejd"https://zbmath.org/authors/?q=ai:hashorva.enkelejd"Kriukov, Nikolai"https://zbmath.org/authors/?q=ai:kriukov.nikolaiSummary: Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical \(d\)-dimensional Brownian risk model (Brm) [\textit{G. A. Delsing} et al., Methodol. Comput. Appl. Probab. 22, No. 3, 927--948 (2020; Zbl 1455.91217)]. From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least \(k\) out of \(d\) components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of \textit{K. Dȩbicki} et al. [J. Appl. Probab. 57, No. 2, 597--612 (2020; Zbl 1464.60031)] and \textit{K. Dȩbicki} et al. [Stochastic Processes Appl. 128, No. 12, 4171--4206 (2018; Zbl 1417.60028)].Optional decomposition of optional supermartingales and applications to filtering and financehttps://zbmath.org/1496.910812022-11-17T18:59:28.764376Z"Abdelghani, Mohamed"https://zbmath.org/authors/?q=ai:abdelghani.mohamed-n"Melnikov, Alexander"https://zbmath.org/authors/?q=ai:melnikov.alexander-vSummary: The classical Doob-Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob-Meyer decomposition of optional supermartingales on \textit{un}usual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.An analytical approximation for convertible bondshttps://zbmath.org/1496.910832022-11-17T18:59:28.764376Z"Goard, Joanna"https://zbmath.org/authors/?q=ai:goard.joanna-mSummary: This paper looks at adapting the method of \textit{A. Medvedev} and \textit{O. Scaillet} [``Pricing American options under stochastic volatility and stochastic interest rates'', J. Financ. Econ. 98, No. 1, 145--159 (2010; \url{doi:10.1016/j.jfineco.2010.03.017})] for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices.ELS pricing and hedging in a fractional Brownian motion environmenthttps://zbmath.org/1496.910862022-11-17T18:59:28.764376Z"Kim, Seong-Tae"https://zbmath.org/authors/?q=ai:kim.seong-tae"Kim, Hyun-Gyoon"https://zbmath.org/authors/?q=ai:kim.hyun-gyoon"Kim, Jeong-Hoon"https://zbmath.org/authors/?q=ai:kim.jeong-hoonSummary: An equity-linked security (ELS) is a debt instrument with several payments and maturities linked to equity markets. This paper is a study of the pricing and hedging of the ELS when the underlying asset price moves in a geometric fractional Brownian motion environment. We develop two different methods for calibrating fractional implied volatility, obtain an empirical result on the Hurst exponent, and introduce a new Greek called \(Eta\) to find the sensitivity of the ELS price to the Hurst parameter. We propose three Delta hedging strategies and compare them with each other and the classical Black-Scholes Delta hedging strategy. Their performance is shown to depend on market circumstance (bull or bear). Our results with constant volatility and Hurst exponent provide a building basis for more stable hedging strategies in the non-Markov environment.Analytical solution of American cull option under fractional Black and Scholes modelhttps://zbmath.org/1496.910892022-11-17T18:59:28.764376Z"Mohamed, Kharrat"https://zbmath.org/authors/?q=ai:mohamed.kharratSummary: The aim of this paper is to give an analytical solution of American call option generated by the fractional Black and Scholes model using the Adomian decomposition method.Dynamic optimal hedge ratio design when price and production are stochastic with jumphttps://zbmath.org/1496.910902022-11-17T18:59:28.764376Z"Nyassoke Titi, Gaston Clément"https://zbmath.org/authors/?q=ai:nyassoke-titi.gaston-clement"Sadefo Kamdem, Jules"https://zbmath.org/authors/?q=ai:sadefo-kamdem.jules"Fono, Louis Aimé"https://zbmath.org/authors/?q=ai:fono.louis-aimeSummary: In this paper, we focus on the farmer's risk income when using commodity futures, when price and output processes are randomly correlated and represented by jump-diffusion models. We evaluate the expected utility of the farmer's wealth and determine the optimal consumption rate and hedging position at each point in time given the harvest timing and state variables. We find a closed form for the optimal consumption and positioning rate in the case of an investor with CARA utility. This result (see Table 3.3) is a generalization of the result of \textit{T. S. Y. Ho} [``Intertemporal commodity futures hedging and the production decision'', J. Finance 39, 351--376 (1984; \url{doi:10.1111/j.1540-6261.1984.tb02314.x})], which considers the special case in which price and output are diffusion models.Pricing of financial derivatives based on the Tsallis statistical theoryhttps://zbmath.org/1496.910932022-11-17T18:59:28.764376Z"Zhao, Pan"https://zbmath.org/authors/?q=ai:zhao.pan"Pan, Jian"https://zbmath.org/authors/?q=ai:pan.jian"Yue, Qin"https://zbmath.org/authors/?q=ai:yue.qin"Zhang, Jinbo"https://zbmath.org/authors/?q=ai:zhang.jinboSummary: Asset return distributions usually have peaks, fat tails and skewed tails, because of the impact of extreme events outside financial markets. The Tsallis distribution has the peak and fat-tail characteristic, and the asymmetric jump process can fit the skewed tail of returns. Therefore, to accurately describe asset returns, we propose a price model by the use of the Tsallis distribution and a Poisson jump process, which can characterize the long-term memory and the skewness of asset returns. Moreover, using the stochastic differential theory and the martingale method, we obtain an explicit solution for pricing European options.Explicit solution simulation method for the 3/2 modelhttps://zbmath.org/1496.910942022-11-17T18:59:28.764376Z"Kouarfate, Iro René"https://zbmath.org/authors/?q=ai:kouarfate.iro-rene"Kouritzin, Michael A."https://zbmath.org/authors/?q=ai:kouritzin.michael-a"MacKay, Anne"https://zbmath.org/authors/?q=ai:mackay.anneSummary: An explicit weak solution for the 3/2 stochastic volatility model is obtained and used to develop a simulation algorithm for option pricing purposes. The 3/2 model is a non-affine stochastic volatility model whose variance process is the inverse of a CIR process. This property is exploited here to obtain an explicit weak solution, similarly to [\textit{M. A. Kouritzin}, Int. J. Theor. Appl. Finance 21, No. 1, Article ID 1850006, 45 p. (2018; Zbl 1395.91454)] for the Heston model. A simulation algorithm based on this solution is proposed and tested via numerical examples. The performance of the resulting pricing algorithm is comparable to that of other popular simulation algorithms.
For the entire collection see [Zbl 1478.60006].Probabilistic prediction of credit ratings: a filtering approachhttps://zbmath.org/1496.910972022-11-17T18:59:28.764376Z"Tardelli, Paola"https://zbmath.org/authors/?q=ai:tardelli.paolaSummary: In a financial market, to analyze the credit quality of firms, this note proposes a dynamical model. The population of firms is divided into a finite number of classes, depending on their credit status. The cardinality of the population can increase during the time, since new firms can enter in the market. Due to changes in credit quality and to the defaults, each firm can move from a class to another, or can go to the class of the defaulted firms. Different rating agencies are considered, each of them defines its own partition of the population. Aim of this note is to find the probabilistic prediction of the actual partition of the population, and of the conditional distribution of the distance to defaults. In a partial observing setting, this topic is discussed using stochastic filtering techniques.Dividends and compound Poisson processes: a new stochastic stock price modelhttps://zbmath.org/1496.910992022-11-17T18:59:28.764376Z"Gankhuu, Battulga"https://zbmath.org/authors/?q=ai:gankhuu.battulga"Kleinow, Jacob"https://zbmath.org/authors/?q=ai:kleinow.jacob"Lkhamsuren, Altangerel"https://zbmath.org/authors/?q=ai:lkhamsuren.altangerel"Horsch, Andreas"https://zbmath.org/authors/?q=ai:horsch.andreasReliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo typehttps://zbmath.org/1496.911012022-11-17T18:59:28.764376Z"Jin, Ting"https://zbmath.org/authors/?q=ai:jin.ting"Ding, Hui"https://zbmath.org/authors/?q=ai:ding.hui"Xia, Hongxuan"https://zbmath.org/authors/?q=ai:xia.hongxuan"Bao, Jinfeng"https://zbmath.org/authors/?q=ai:bao.jinfengSummary: This paper mainly studies the pricing problem of arithmetic average Asian barrier options in the continuous case and analyzes the corresponding reliability index. Owing to the fact that the return function of the option is closely related to the average price of the underlying asset in a certain period, which can effectively alleviate the market speculation, the Asian barrier option is widely active in the financial market as a derivative product. Firstly, considering that there exists difficult to predict and measure reliability based on historical data, traditional stock models relied on stochastic theory are abandoned, and further assumes that the underlying assets follow an uncertain process. Then, we introduce the uncertain fractional differential equations with Caputo-type to describe the dynamic changes of risky asset prices. Secondly, in order to analyze the impact on the first hitting time of the barrier being triggered and the option execution result theoretically, a novel first-hitting time model is established to measure the relationship between the option reliability index and the value of risky assets. Meanwhile, the pricing formulas of four Asian barrier options and the corresponding reliability index under the first-hitting time model are derived, respectively. Finally, numerical algorithms and corresponding methods are designed to verify the results of our model.Applying the local martingale theory of bubbles to cryptocurrencieshttps://zbmath.org/1496.911022022-11-17T18:59:28.764376Z"Choi, Soon Hyeok"https://zbmath.org/authors/?q=ai:choi.soon-hyeok"Jarrow, Robert A."https://zbmath.org/authors/?q=ai:jarrow.robert-aA PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulationshttps://zbmath.org/1496.920152022-11-17T18:59:28.764376Z"Darrigade, Léo"https://zbmath.org/authors/?q=ai:darrigade.leo"Haghebaert, Marie"https://zbmath.org/authors/?q=ai:haghebaert.marie"Cherbuy, Claire"https://zbmath.org/authors/?q=ai:cherbuy.claire"Labarthe, Simon"https://zbmath.org/authors/?q=ai:labarthe.simon"Laroche, Beatrice"https://zbmath.org/authors/?q=ai:laroche.beatriceSummary: Human health and physiology is strongly influenced by interactions between human cells and intestinal microbiota in the gut. In mammals, the host-microbiota crosstalk is mainly mediated by regulations at the intestinal crypt level: the epithelial cell turnover in crypts is directly influenced by metabolites produced by the microbiota. Conversely, enterocytes maintain hypoxia in the gut, favorable to anaerobic bacteria which dominate the gut microbiota. We constructed an individual-based model of epithelial cells interacting with the microbiota-derived chemicals diffusing in the crypt lumen. This model is formalized as a piecewise deterministic Markov process (PDMP). It accounts for local interactions due to cell contact (among which are mechanical interactions), for cell proliferation, differentiation and extrusion which are regulated spatially or by chemicals concentrations. It also includes chemicals diffusing and reacting with cells. A deterministic approximated model is also introduced for a large population of small cells, expressed as a system of porous media type equations. Both models are extensively studied through numerical exploration. Their biological relevance is thoroughly assessed by recovering bio-markers of an healthy crypt, such as cell population distribution along the crypt or population turn-over rates. Simulation results from the deterministic model are compared to the PMDP model and we take advantage of its lower computational cost to perform a sensitivity analysis by Morris method. We finally use the crypt model to explore butyrate supplementation to enhance recovery after infections by enteric pathogens.Is the Allee effect relevant to stochastic cancer model?https://zbmath.org/1496.920372022-11-17T18:59:28.764376Z"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhas(no abstract)Bistability in deterministic and stochastic SLIAR-type models with imperfect and waning vaccine protectionhttps://zbmath.org/1496.920412022-11-17T18:59:28.764376Z"Arino, Julien"https://zbmath.org/authors/?q=ai:arino.julien"Milliken, Evan"https://zbmath.org/authors/?q=ai:milliken.evanSummary: Various vaccines have been approved for use to combat COVID-19 that offer imperfect immunity and could furthermore wane over time. We analyze the effect of vaccination in an SLIARS model with demography by adding a compartment for vaccinated individuals and considering disease-induced death, imperfect and waning vaccination protection as well as waning infections-acquired immunity. When analyzed as systems of ordinary differential equations, the model is proven to admit a backward bifurcation. A continuous time Markov chain (CTMC) version of the model is simulated numerically and compared to the results of branching process approximations. While the CTMC model detects the presence of the backward bifurcation, the branching process approximation does not. The special case of an SVIRS model is shown to have the same properties.Pedigree in the biparental Moran modelhttps://zbmath.org/1496.920502022-11-17T18:59:28.764376Z"Coron, Camille"https://zbmath.org/authors/?q=ai:coron.camille"Le Jan, Yves"https://zbmath.org/authors/?q=ai:le-jan.yvesThis paper is concerned with the genetic composition of a population with biparental genetic transmission. The authors focus on the Moran biparental model, in which 1) each individual has 2 parents, who contribute equally to the genome of their offspring, 2) both parents and the replaced individual are in different sites and 3) the number of individuals is fixed, given by \(N\). The weight of an ancestor is defined to be the probability that a given gene of the sampled individual comes from this ancestor. Under this idealized model, the authors first show that the weights of an ancestor in all individuals are asymptotically equal. As \(N\) goes to infinity, the total weights of \(l\) ancestors in the population (properly rescaled by a factor of \(N\)) converge in law to a vector of \(l\) independent random variables, that are the mixture of a Dirac measure in 0 and an exponential law with parameter 1/2.
The key points of the proofs are 1) introducing the pedigree graph of the population, 2) Considering the stationary distribution of a \(k\)-particle random walk on the random pedigree graph and 3) a certain projection of this \(k\)-particles random walk remains a Markov chain.
Reviewer: Thomas Jiaxian Li (Charlottesville)Stability of three species symbiosis model with delay and stochastic perturbationshttps://zbmath.org/1496.920792022-11-17T18:59:28.764376Z"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed"Shaikhet, Leonid"https://zbmath.org/authors/?q=ai:shaikhet.leonid-eSummary: In this paper a three-species symbiosis population model with delay and stochastic perturbations is considered. The model is modified by considering more general rate which ensure the existence of at least one nontrivial equilibrium. Conditions for the existence of positive equilibrium of the considered model are obtained. New sufficient conditions of stability in probability for the obtained positive equilibrium are formulated in the terms of linear matrix inequalities (LMIs), which can be investigated by virtue of MATLAB. Besides some necessary stability conditions are formulated in the form of simple analytical inequalities. The results obtained are illustrated via numerical simulations of a solution of the considered model.Stochastic predator-prey Lévy jump model with Crowley-Martin functional response and stage structurehttps://zbmath.org/1496.920882022-11-17T18:59:28.764376Z"Danane, Jaouad"https://zbmath.org/authors/?q=ai:danane.jaouadThe author modifies the predator-prey model with mature and immature of predator and prey,
\begin{align*}
\frac{dx_1}{dt}&=ax_2-d_1x_1-px_1 \\
\frac{dx_2}{dt}&=px_1-d_2x_2-b_1x_2^2-\frac{cx_2y_2}{1+\alpha x_2+\beta y_2+\alpha\beta x_2y_2} \\
\frac{dy_1}{dt}&=\frac{ecx_2y_2}{1+\alpha x_2+\beta y_2+\alpha\beta x_2y_2}-d_3y_1-hy_1 \\
\frac{dy_2}{dt}&=hy_1-d_4y_2-b_2y_2^2,
\end{align*}
and introduces a stochastic model with Lévy noise, using the Crowley-Martin functional response. White noise is also incorporated in each model compartment. Existence of the unique solution global in time is confirmed, and then several conditions on the parameter, concerning the extinction of both prey and predator, extinction of predator and persistent of prey, and persistent of both prey and predator, are derived.
Reviewer: Takashi Suzuki (Osaka)Stability in distribution of a stochastic predator-prey system with S-type distributed time delayshttps://zbmath.org/1496.920982022-11-17T18:59:28.764376Z"Wang, Sheng"https://zbmath.org/authors/?q=ai:wang.sheng"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixin"Wei, Tengda"https://zbmath.org/authors/?q=ai:wei.tengda"Wang, Linshan"https://zbmath.org/authors/?q=ai:wang.linshanThe paper studies stability of the following Lotka-Volterra stochastic system
\[
d x_1(t)=x_1(t)\left(r_1-a_{11}x_1(t)-\int_{-\tau_{11}}^0 x_1(t+\theta) d \mu_{11}(\theta)-a_{12}x_2(t)\right.
\]
\[
\left.-\int_{-\tau_{12}}^0 x_2(t+\theta) d \mu_{12}(\theta)\right)dt+\sigma_1x_1(t)d B_1(t)
\]
\[
d x_2(t)=x_2(t)\left(r_2-a_{22}x_2(t)-\int_{-\tau_{22}}^0 x_2(t+\theta) d \mu_{22}(\theta)-a_{21}x_1(t)\right.
\]
\[
\left.-\int_{-\tau_{21}}^0 x_1(t+\theta) d \mu_{21}(\theta)\right)dt+\sigma_2x_2(t)d B_2(t),
\]
where \(B_i(t)\) are mutually independent standard Wiener processes, \(\tau_{ij}>0\).
Reviewer: Leonid Berezanski (Be'er Sheva)Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switchinghttps://zbmath.org/1496.920992022-11-17T18:59:28.764376Z"Wang, Zhaojuan"https://zbmath.org/authors/?q=ai:wang.zhaojuan"Deng, Meiling"https://zbmath.org/authors/?q=ai:deng.meiling"Liu, Meng"https://zbmath.org/authors/?q=ai:liu.mengSummary: This article explores a stochastic ratio-dependent predator-prey model with regime-switching. We testify that the model admits a unique stationary distribution, and demonstrate that the transition probability of the solution of the model converges to the stationary distribution in exponent rate. We also discuss the biological implications of the results by aid of some numerical simulations.Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switchinghttps://zbmath.org/1496.921042022-11-17T18:59:28.764376Z"Boukanjime, Brahim"https://zbmath.org/authors/?q=ai:boukanjime.brahim"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"El Fatini, Mohamed"https://zbmath.org/authors/?q=ai:el-fatini.mohamed"El Khalifi, Mohamed"https://zbmath.org/authors/?q=ai:el-khalifi.mohamedSummary: In this paper, we analyze a stochastic coronavirus (COVID-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. Firstly, we investigate the existence and uniqueness of a global positive solution. Then, we establish the stochastic threshold for the extinction and the persistence of the disease. The data from Indian states, are used to confirm the results established along this paper.Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growthhttps://zbmath.org/1496.921142022-11-17T18:59:28.764376Z"Han, Bingtao"https://zbmath.org/authors/?q=ai:han.bingtao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqing"Zhou, Baoquan"https://zbmath.org/authors/?q=ai:zhou.baoquan"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawar"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmedSummary: Focusing on the results of [\textit{S. P. Rajasekar} and \textit{M. Pitchaimani}, Appl. Math. Comput. 377, Article ID 125143, 15 p. (2020; Zbl 1488.92076)] and the continuous dynamics of stochastic differential equation (SDE) developed by \textit{X. Mao} [Stochastic differential equations and their applications. Chichester: Horwood Publishing (1997; Zbl 0892.60057)], a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is investigated in this paper. First, we propose and prove that the unique solution of stochastic model is globally positive. By constructing some suitable Lyapunov functions, the sufficient condition \(\mathcal{R}_0^h>1\) is obtained for the unique stationary distribution which has ergodicity property. Next, by solving the corresponding Fokker-Planck equation, we derive the approximate probability density function around the quasi-endemic equilibrium of the stochastic system. The above stationary distribution and density function can reveal all statistical properties of the disease persistence. In addition, by comparison with other existing articles, our developed theoretical results and some numerical simulations are introduced at the end of this paper.A stochastic SIR epidemic evolution model with non-concave force of infection: mathematical modeling and analysishttps://zbmath.org/1496.921172022-11-17T18:59:28.764376Z"Lahrouz, A."https://zbmath.org/authors/?q=ai:lahrouz.aadil"Settati, A."https://zbmath.org/authors/?q=ai:settati.adel"Jarroudi, M."https://zbmath.org/authors/?q=ai:jarroudi.m"Mahjour, H."https://zbmath.org/authors/?q=ai:mahjour.h"Fatini, M."https://zbmath.org/authors/?q=ai:fatini.m-el"Merzguioui, M."https://zbmath.org/authors/?q=ai:merzguioui.m"Tridane, A."https://zbmath.org/authors/?q=ai:tridane.abdessamadItô vs Stratonovich stochastic SIR modelshttps://zbmath.org/1496.921182022-11-17T18:59:28.764376Z"Lanconelli, Alberto"https://zbmath.org/authors/?q=ai:lanconelli.alberto"Mori, Matteo"https://zbmath.org/authors/?q=ai:mori.matteoSummary: We prove that the asymptotic behavior of the Stratonovich counterpart of the Itô's type stochastic SIR model investigated by \textit{E. Tornatore} et al. [``Stability of a stochastic SIR system'', Physica A, 354, 111--126 (2005; \url{doi:10.1016/j.physa.2005.02.057})], and \textit{C. Ji} and \textit{D. Jiang} [Appl. Math. Modelling 38, No. 21--22, 5067--5079 (2014; Zbl 1428.92109); \textit{C. Ji} et al., Stochastic Anal. Appl. 30, No. 5, 755--773 (2012; Zbl 1272.60035)] is ruled by the same threshold as the deterministic system. In other words, in contrast to the Itô's model, the intensity of the noise described through the Stratonovich calculus is not relevant for the extinction of the disease. The Stratonovich interpretation of the model is motivated by the parameter perturbation technique, employed on the disease transmission coefficient and used to implement environmental randomness, in combination with the classical Wong-Zakai approximation argument.Dynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and ameliorationhttps://zbmath.org/1496.921192022-11-17T18:59:28.764376Z"Liu, Qun"https://zbmath.org/authors/?q=ai:liu.qun"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqingSummary: In this paper, we study the dynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration. We derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate criteria for complete eradication and wiping out of the infectious disease. Finally, we introduce some numerical simulations to confirm the theoretical results.Stochastic analysis of a SIRI epidemic model with double saturated rates and relapsehttps://zbmath.org/1496.921262022-11-17T18:59:28.764376Z"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.5"Gao, Shujing"https://zbmath.org/authors/?q=ai:gao.shujing"Chen, Shihua"https://zbmath.org/authors/?q=ai:chen.shihuaSummary: Infectious diseases have for centuries been the leading causes of death and disability worldwide and the environmental fluctuation is a crucial part of an ecosystem in the natural world. In this paper, we proposed and discussed a stochastic SIRI epidemic model incorporating double saturated incidence rates and relapse. The dynamical properties of the model were analyzed. The existence and uniqueness of a global positive solution were proven. Sufficient conditions were derived to guarantee the extinction and persistence in mean of the epidemic model. Additionally, ergodic stationary distribution of the stochastic SIRI model was discussed. Our results indicated that the intensity of relapse and stochastic perturbations greatly affected the dynamics of epidemic systems and if the random fluctuations were large enough, the disease could be accelerated to extinction while the stronger relapse rate were detrimental to the control of the disease.A classification of the dynamics of three-dimensional stochastic ecological systemshttps://zbmath.org/1496.921292022-11-17T18:59:28.764376Z"Hening, Alexandru"https://zbmath.org/authors/?q=ai:hening.alexandru"Nguyen, Dang H."https://zbmath.org/authors/?q=ai:nguyen.dang-hai"Schreiber, Sebastian J."https://zbmath.org/authors/?q=ai:schreiber.sebastian-jAuthors' abstract: The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis' conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis' conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka-Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology's modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.
Reviewer: Yuming Chen (Waterloo)Asymptotic diffusion analysis of \(MMPP|M|N\) queueing systems with feedbackhttps://zbmath.org/1496.930462022-11-17T18:59:28.764376Z"Nazarov, A. A."https://zbmath.org/authors/?q=ai:nazarov.anatoly-a|nazarov.anton-a"Pavlova, E. A."https://zbmath.org/authors/?q=ai:pavlova.e-a.1Summary: We present the results of the study of an \(N \)-linear queueing system with feedback. The input flow is a Markov modulated Poisson process (MMPP). We use asymptotic diffusion analysis to find the probability distribution of the number of requests in the orbit and the number of busy servers in the system. Simulation results, as well as a numerical comparison of our method with asymptotic analysis method, are presented.Existence of relaxed optimal control for \(G\)-neutral stochastic functional differential equations with uncontrolled diffusionhttps://zbmath.org/1496.931272022-11-17T18:59:28.764376Z"Elgroud, Nabil"https://zbmath.org/authors/?q=ai:elgroud.nabil"Boutabia, Hacene"https://zbmath.org/authors/?q=ai:boutabia.hacene"Redjil, Amel"https://zbmath.org/authors/?q=ai:redjil.amel"Kebiri, Omar"https://zbmath.org/authors/?q=ai:kebiri.omarSummary: In this paper, we study under refined Lipschitz hypothesis, the question of existence and uniqueness of solution of controlled neutral stochastic functional differential equations driven by \(G\)-Brownian motion \((G\)-NSFDEs in short). An existence of a relaxed optimal control where the neutral and diffusion terms do not depend on the control variable was the main result of the article. The latter is done by using tightness techniques and the weak convergence techniques for each probability measure in the set of all possible probabilities of our dynamic. A motivation of our work is presented and a numerical analysis for the uncontrolled \(G\)-NSFDE is given.