Recent zbMATH articles in MSC 60Ghttps://zbmath.org/atom/cc/60G2023-09-19T14:22:37.575876ZWerkzeugOn the convergence scheme in the CRR modelhttps://zbmath.org/1516.050232023-09-19T14:22:37.575876Z"Kostrzewa, Tomasz"https://zbmath.org/authors/?q=ai:kostrzewa.tomaszSummary: We investigate the convergence scheme from the discrete to the continuous time model in the binomial tree model of Cox-Ross-Rubinstein (CRR). We introduce the notion of \(\bar\Sigma\)-decomposition and we classify the financial payoffs according to their representation in terms of \(\bar\Sigma\) functions in the CRR model. We find the exact convergence rate for a single \(\bar\Sigma\) function and we obtain the general convergence scheme for their linear combinations. This way, we get a universal and convenient tool to investigate the payoffs in the continuous time model by reducing the problem to the discrete time model. An illustration of this method is retrieving the formula for the double barrier option price in the Black-Scholes model, in the form of the sum of an infinite series, derived by \textit{N. Kunitomo} and \textit{M. Ikeda} in 1992 [Math. Finance 2, No. 4, 275--298 (1992; Zbl 0900.90098)]. To the best of our knowledge, it is the first time that the double barrier options payoff is obtained by convergence from the discrete time model. Finally, we show that an alternative form of the results obtained corresponds to \textit{P. Biane}'s research in the area of the Riemann function [``La fonction zêta de Riemann et les probabilités'', in: La fonction zêta. Palaiseau: Les Éditions de l'École Polytechnique (2003), contained in Zbl 1026.11069].Triangulations of uniform subquadratic growth are quasi-treeshttps://zbmath.org/1516.050412023-09-19T14:22:37.575876Z"Benjamini, Itai"https://zbmath.org/authors/?q=ai:benjamini.itai"Georgakopoulos, Agelos"https://zbmath.org/authors/?q=ai:georgakopoulos.agelosSummary: It is known that for every \(\alpha \geqslant 1\) there is a planar triangulation in which every ball of radius \(r\) has size \(\Theta(r^\alpha)\). We prove that for \(\alpha < 2\) every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree.An Itô calculus for a class of limit processes arising from random walks on the complex planehttps://zbmath.org/1516.351802023-09-19T14:22:37.575876Z"Bonaccorsi, Stefano"https://zbmath.org/authors/?q=ai:bonaccorsi.stefano"Calcaterra, Craig"https://zbmath.org/authors/?q=ai:calcaterra.craig"Mazzucchi, Sonia"https://zbmath.org/authors/?q=ai:mazzucchi.soniaSummary: Within the framework of the previous paper [\textit{S. Bonaccorsi} and \textit{S. Mazzucchi}, Stochastic Processes Appl. 125, No. 2, 797--818 (2015; Zbl 1323.35225)], we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.Approximate normality of high-energy hyperspherical eigenfunctionshttps://zbmath.org/1516.352882023-09-19T14:22:37.575876Z"Campese, Simon"https://zbmath.org/authors/?q=ai:campese.simon"Marinucci, Domenico"https://zbmath.org/authors/?q=ai:marinucci.domenico"Rossi, Maurizia"https://zbmath.org/authors/?q=ai:rossi.mauriziaSummary: The Berry heuristic has been a long standing \textit{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see [\textit{M. V. Berry}, J. Phys. A, Math. Gen. 10, 2083--2091 (1977; Zbl 0377.70014)]). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized \(d\)-dimensional sphere -- also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.Gaussian process hydrodynamicshttps://zbmath.org/1516.353102023-09-19T14:22:37.575876Z"Owhadi, H."https://zbmath.org/authors/?q=ai:owhadi.houman|owhadi.howman(no abstract)Dissipative solutions and Markov selection to the complete stochastic Euler systemhttps://zbmath.org/1516.353182023-09-19T14:22:37.575876Z"Moyo, Thamsanqa Castern"https://zbmath.org/authors/?q=ai:moyo.thamsanqa-casternSummary: We introduce the concept of \textit{stochastic measure-valued solutions} to the complete Euler system describing the motion of a compressible inviscid fluid subject to stochastic forcing, where the nonlinear terms are described by defect measures. These solutions are weak in the probabilistic sense (probability space is not a given `priori', but part of the solution) and analytical sense (derivatives only exist in the sense distributions). In particular, we show that existence and weak-strong principle (i.e. a weak measure-valued solution coincides with a strong solution provided the later exists) hold true provided they satisfy some form of energy balance. Finally, we show the existence of Markov selection to the associated martingale problem.Optimal investment with stopping in finite horizonhttps://zbmath.org/1516.355532023-09-19T14:22:37.575876Z"Jian, Xiongfei"https://zbmath.org/authors/?q=ai:jian.xiongfei"Li, Xun"https://zbmath.org/authors/?q=ai:li.xun"Yi, Fahuai"https://zbmath.org/authors/?q=ai:yi.fahuaiSummary: In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature. We formulate our model to a free boundary problem of a fully \textit{nonlinear} equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a \textit{linear} equation. Finally, we apply the theoretical results to some challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon.The spatial \(\Lambda\)-Fleming-Viot process in a random environmenthttps://zbmath.org/1516.355762023-09-19T14:22:37.575876Z"Klimek, Aleksander"https://zbmath.org/authors/?q=ai:klimek.aleksander"Rosati, Tommaso Cornelis"https://zbmath.org/authors/?q=ai:rosati.tommaso-cornelisSummary: We study the large scale behaviour of a population consisting of two types which evolve in dimension \(d=1,2\) according to a spatial Lambda-Fleming-Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher-KPP equation in a random potential. The proofs build on two-scale Schauder estimates and semidiscrete approximations of the Anderson Hamiltonian.Invariant measures for SDEs driven by Lévy noise: a case study for dissipative nonlinear drift in infinite dimensionhttps://zbmath.org/1516.355792023-09-19T14:22:37.575876Z"Albeverio, Sergio"https://zbmath.org/authors/?q=ai:albeverio.sergio-a"Di Persio, Luca"https://zbmath.org/authors/?q=ai:di-persio.luca"Mastrogiacomo, Elisa"https://zbmath.org/authors/?q=ai:mastrogiacomo.elisa"Smii, Boubaker"https://zbmath.org/authors/?q=ai:smii.boubakerSummary: We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. We define a Hilbert-Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process into a stationary component, the law of which is identified with the unique invariant probability measure \(\mu\) of the process, and a component which vanishes asymptotically for large times in the \(L^p(\mu)\)-sense, for all \(1\leq p < +\infty\).Symmetric \(\alpha\)-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudodifferential equationhttps://zbmath.org/1516.355852023-09-19T14:22:37.575876Z"Osypchuk, M. M."https://zbmath.org/authors/?q=ai:osypchuk.m-m"Portenko, M. I."https://zbmath.org/authors/?q=ai:portenko.mykola-ivanovychSummary: We consider a pseudodifferential equation of parabolic type with operator of fractional differentiation with respect to a space variable generating a symmetric \(\alpha\)-stable process in a multidimensional Euclidean space with initial and boundary conditions imposed on the values of an unknown function at points of the boundary of a given domain. The imposed boundary condition is similar to the condition of the so-called third (mixed) boundary-value problem in the theory of differential equations with the sole difference that the traditional (co)normal derivative is replaced in our problem with a pseudodifferential operator. Another specific feature of the analyzed problem is the two-sided character of the boundary condition obtained as a consequence of the fact that, in the case of \(\alpha\) with values between 1 and 2, the corresponding process reaches the boundary and makes infinitely many visits to both the interior and exterior regions with respect to the boundary.Asymptotic problems on homogeneous spaceshttps://zbmath.org/1516.370022023-09-19T14:22:37.575876Z"Södergren, Anders"https://zbmath.org/authors/?q=ai:sodergren.andersSummary: This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces.
In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution.
Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians.
In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.Generic non-singular Poisson suspension is of type \(\mathrm{III}_1\)https://zbmath.org/1516.370102023-09-19T14:22:37.575876Z"Danilenko, Alexandre I."https://zbmath.org/authors/?q=ai:danilenko.alexandre-i"Kosloff, Zemer"https://zbmath.org/authors/?q=ai:kosloff.zemer"Roy, Emmanuel"https://zbmath.org/authors/?q=ai:roy.emmanuelSummary: It is shown that for a dense \(G_\delta\)-subset of the subgroup of non-singular transformations (of a standard infinite \(\sigma\)-finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger's type \(\mathrm{III}_1\).Exponential and strong ergodicity for one-dimensional time-changed symmetric stable processeshttps://zbmath.org/1516.370112023-09-19T14:22:37.575876Z"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao.34The author studies ergodic properties, such as exponential ergodicity and strong ergodicity, of the weak solution of the stochastic differential equation
\[
dY_t=\sigma (Y_{t-})dX_t,
\]
where \(\sigma \) is a strictly positive continuous function on \(\mathbb{R}\) and \(X = (X_t)_{t\geq 0}\) is a symmetric \(\alpha\)-stable process on \(\mathbb{R}\) with an infinitesimal generator \(\Delta ^{\alpha /2}\) with \(0< \alpha < 2\). For \(\alpha \in (1,2)\), since \(X\) is pointwise recurrent, it is known that \(Y\) is ergodic if the measure \(\mu \) is finite on \(\mathbb{R}\), where \(\mu (dx) := \sigma (x)^{-\alpha }dx\).
The author further investigates ergodic properties of \(Y\) and provides explicit criteria for exponential ergodicity and strong ergodicity. It is shown (Theorem 1.1) that \(Y\) is exponentially ergodic if and only if the quantity
\[
\sup _{x}|x|^{\alpha -1}\int _{\mathbb{R}\backslash (-|x|,|x|)}\sigma (y)^{-\alpha }dy
\]
is finite.
On the other hand, it is shown (Theorem 1.4) that \(Y\) is strongly ergodic if and only if the quantity
\[
\int _{\mathbb{R}}\sigma (x)^{-\alpha }|x|^{\alpha -1}dx
\]
is finite.
Reviewer: Azer Akhmedov (Fargo)On the action of multiplicative cascades on measureshttps://zbmath.org/1516.370582023-09-19T14:22:37.575876Z"Barral, Julien"https://zbmath.org/authors/?q=ai:barral.julien"Jin, Xiong"https://zbmath.org/authors/?q=ai:jin.xiongSummary: We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than Hölder), all our results are sharp. This improves results previously obtained by \textit{J. P. Kahane} and \textit{J. Peyriere} [Adv. Math. 22, 131--145 (1976; Zbl 0349.60051)], \textit{F. Ben Nasr} [C. R. Acad. Sci., Paris, Sér. I 304, 255--258 (1987; Zbl 0635.60052)], and \textit{A. Fan} [Stud. Math. 111, No. 1, 1--17 (1994; Zbl 0805.28002)]. We exploit our results to derive dimension estimates and absolute continuity for some random fractal measures.Martingale inequalities for spline sequenceshttps://zbmath.org/1516.410072023-09-19T14:22:37.575876Z"Passenbrunner, Markus"https://zbmath.org/authors/?q=ai:passenbrunner.markusSummary: We show that D. Lépingle's \(L_1(\ell_2)\)-inequality \[ \left\| \left( \sum_n{\mathbb{E}}[f_n | \mathscr{F}_{n-1}]^2 \right)^{1/2}\right\|_1 \le 2\cdot \left\| \left( \sum_n f_n^2 \right)^{1/2} \right\|_1, \quad f_n\in\mathscr{F}_n, \] extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that \(f_n\) is contained in a suitable spline space \(\mathscr{S}(\mathscr{F}_n)\). This is done provided the filtration \((\mathscr{F}_n)\) satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in \(\mathscr{S}(\mathscr{F}_n)\). As a by-product, we also obtain a spline version of \(H_1\)-BMO duality under this assumption.Markovianity and the Thompson monoid \(F^+\)https://zbmath.org/1516.460422023-09-19T14:22:37.575876Z"Köstler, Claus"https://zbmath.org/authors/?q=ai:kostler.claus"Krishnan, Arundhathi"https://zbmath.org/authors/?q=ai:krishnan.arundhathi"Wills, Stephen J."https://zbmath.org/authors/?q=ai:wills.stephen-jThe paper under review develops a new notion of symmetry, the so-called partial spreadability, for unilateral non-commutative (but also classical) stochastic processes. The starting point of the study is the observation made in [\textit{D.~G. Evans} et al., Rocky Mt. J. Math. 47, No.~6, 1839--1873 (2017; Zbl 1401.18039)], saying that the usual spreadability is related to existence of a suitable representation of a certain `shift monoid' \(S^+\) via endomorphism of the associated noncommutative (or classical) probability space. Partial spreadability is a weakening of this condition to the existence of an analogous representation of a Thompson monoid, \(F^+\). The authors show that partially spreadable processes are stationary (and adapted to a local Markov filtration); and in the commutative case under a natural maximality condition, partial spreadability is in fact equivalent to Markovianity and stationarity. The proof of the latter fact is based on the existence of tensor dilations, as established in the habilitation thesis of Burkhard Kümmerer.
The paper presents the background results in full detail, and treats the commutative case practically separately from the general situation. This makes it self-contained and useful to readers of different backgrounds, but, perhaps neccessarily, forces the authors to repeat certain results several times over the course of the article.
Reviewer: Adam Skalski (Warszawa)Existence and uniqueness for variational data assimilation in continuous timehttps://zbmath.org/1516.490182023-09-19T14:22:37.575876Z"Bröcker, Jochen"https://zbmath.org/authors/?q=ai:brocker.jochenSummary: A variant of the optimal control problem is considered which is nonstandard in that the performance index contains ``stochastic'' integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be ``fitted'' with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.On the mixing time of coordinate hit-and-runhttps://zbmath.org/1516.520072023-09-19T14:22:37.575876Z"Narayanan, Hariharan"https://zbmath.org/authors/?q=ai:narayanan.hariharan"Srivastava, Piyush"https://zbmath.org/authors/?q=ai:srivastava.piyushSummary: We obtain a polynomial upper bound on the mixing time \(T_{CHR}(\epsilon)\) of the coordinate Hit-and-Run (CHR) random walk on an \(n\)-dimensional convex body, where \(T_{CHR}(\epsilon)\) is the number of steps needed to reach within \(\epsilon\) of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in \(n\), \(R\) and \(\frac{1}{\epsilon}\), where we assume that the convex body contains the unit \(\Vert\cdot\Vert_\infty\)-unit ball \(B_\infty\) and is contained in its \(R\)-dilation \(R\cdot B_\infty\). Whether CHR has a polynomial mixing time has been an open question.Simplified calculus for semimartingales: multiplicative compensators and changes of measurehttps://zbmath.org/1516.600092023-09-19T14:22:37.575876Z"Černý, Aleš"https://zbmath.org/authors/?q=ai:cerny.ales"Ruf, Johannes"https://zbmath.org/authors/?q=ai:ruf.johannesSummary: The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation when such compensation is meaningful. This generalization of the Lévy-Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a signed stochastic exponential, which in turn has practical applications in mean-variance portfolio theory. Girsanov-type results based on multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the characteristic function of log returns for a popular class of minimax measures in a Lévy setting.Generalization of the Glivenko-Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fieldshttps://zbmath.org/1516.600102023-09-19T14:22:37.575876Z"Tempelman, Arkady"https://zbmath.org/authors/?q=ai:tempelman.arkady-aSummary: Let \(X(t), t \in \mathbb{Z}^m \times \mathbb{R}^k\), be an ergodic stationary random processes or an ergodic homogeneous random field \((k+m \geq 1)\). We prove that the distribution function of each random vector \((X(t_1), \dots, X(t_s))\) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets \(T_n\) with infinitely increasing intrinsic diameters are used. We consider also the case when \(X\) is observed on a mixing homogeneous countable random set \(S^m (\omega) \subset \mathbb{R}^m\) (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.Hoffmann-Jørgensen inequalities for random walks on the cone of positive definite matriceshttps://zbmath.org/1516.600112023-09-19T14:22:37.575876Z"Bagyan, Armine"https://zbmath.org/authors/?q=ai:bagyan.armine"Richards, Donald"https://zbmath.org/authors/?q=ai:richards.donald-l|richards.donald-st-pLet \(\mathcal{P}_m\) denote the cone of \(m\times m\) positive definite symmetric matrices, and \(\{X_n:n\in\mathbb{N}\}\) be a sequence of independent and orthogonally invariant random matrices in \(\mathcal{P}_m\). Define the random walk \(S_1=X_1\) and \(S_{j+1}=S_j^{1/2}X_{j+1}S_j^{1/2}\) for \(j=1,2,\dots\) Proximity of elements of \(\mathcal{P}_m\) is measured by the Riemannian metric defined by
\[
d_R(A,B)=\left(\sum_{j=1}^m\left[\log\lambda_j(A^{-1/2}BA^{-1/2})\right]^2\right)^{1/2},
\]
for \(A,B\in\mathcal{P}_m\), with \(\lambda_1(A),\dots,\lambda_m(A)\) denoting the eigenvalues of \(A\).
The authors establish Hoffmann-Jørgensen inequalities for this random walk \(\{S_n:n\in\mathbb{N}\}\). That is, they give upper bounds on the right tail probabilities of \(U_n\), where
\[
U_n=\max\{d_R(I_m,S_1),\dots,d_R(I_m,S_n)\}\,,
\]
and \(I_m\) is the \(m\times m\) identity matrix.
Particular attention is paid to the situation where the \(X_n\) have the Wishart distribution \(W_m(a,I_m)\), i.e., in the case where the underlying random matrices have the following density with respect to Lebesgue measure on \(\mathcal{P}_m\):
\[
w(x)=\frac{1}{2^{ma}\Gamma_m(a)}\left(\det x\right)^{a-\frac{1}{2}(m+1)}\exp\left\{-\frac{1}{2}\text{tr }x\right\}\,,
\]
where \(\Gamma_m(a)=\pi^{m(m-1)/4}\prod_{j=1}^m\Gamma\left(a-\frac{1}{2}(j-1)\right)\) for \(a>\frac{1}{2}(m-1)\), and where \(\Gamma(\cdot)\) is the gamma function. In this example, explicit and computable upper bounds are given for each of the terms appearing in the corresponding Hoffmann-Jørgensen inequality.
Reviewer: Fraser Daly (Edinburgh)On the \(\Phi \)-stability and related conjectureshttps://zbmath.org/1516.600142023-09-19T14:22:37.575876Z"Yu, Lei"https://zbmath.org/authors/?q=ai:yu.leiSummary: Given a convex function \(\Phi :[0,1]\rightarrow{\mathbb{R}}\) and the mean \({\mathbb{E}}f({\mathbf{X}})=a\in [0,1]\), which Boolean function \(f\) maximizes the \(\Phi \)-stability \({\mathbb{E}}[\Phi (T_{\rho }f({\mathbf{X}}))]\) of \(f\)? Here \({\mathbf{X}}\) is a random vector uniformly distributed on the discrete cube \(\{-1,1\}^n\) and \(T_{\rho }\) is the Bonami-Beckner operator. Special cases of this problem include the (symmetric and asymmetric) \( \alpha \)-stability problems and the ``Most Informative Boolean Function'' problem. In this paper, we provide several upper bounds for the maximal \(\Phi \)-stability. When specializing \(\Phi\) to some particular forms, by these upper bounds, we partially resolve Mossel and O'Donnell's conjecture on \(\alpha \)-stability with \(\alpha >2\), Li and Médard's conjecture on \(\alpha \)-stability with \(1<\alpha <2\), and Courtade and Kumar's conjecture on the ``Most Informative Boolean Function'' which corresponds to a conjecture on \(\alpha \)-stability with \(\alpha =1\). Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut-Kalai-Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases.Inheritance of strong mixing and weak dependence under renewal samplinghttps://zbmath.org/1516.600152023-09-19T14:22:37.575876Z"Brandes, Dirk-Philip"https://zbmath.org/authors/?q=ai:brandes.dirk-philip"Curato, Imma Valentina"https://zbmath.org/authors/?q=ai:curato.imma-valentina"Stelzer, Robert"https://zbmath.org/authors/?q=ai:stelzer.robertSummary: Let \(X\) be a continuous-time strongly mixing or weakly dependent process and let \(T\) be a renewal process independent of \(X\). We show general conditions under which the sampled process \((X_{T_i}, T_i - T_{i-1})^{\top}\) is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of \(X\) is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process \(X\) are at our disposal.Central and noncentral limit theorems arising from the scattering transform and its neural activation generalizationhttps://zbmath.org/1516.600162023-09-19T14:22:37.575876Z"Liu, Gi-Ren"https://zbmath.org/authors/?q=ai:liu.gi-ren"Sheu, Yuan-Chung"https://zbmath.org/authors/?q=ai:sheu.yuan-chung"Wu, Hau-Tieng"https://zbmath.org/authors/?q=ai:wu.hau-tiengSummary: Motivated by the analysis of complicated time series, we examine a generalization of the scattering transform that includes broad neural activation functions. This generalization is the \textit{neural activation scattering transform} (NAST). NAST comprises a sequence of ``neural processing units,'' each of which applies a high pass filter to the input from the previous layer followed by a composition with a nonlinear function as the output to the next neuron. Here, the nonlinear function models how a neuron gets excited by the input signal. In addition to showing properties like nonexpansion, horizontal translational invariability, and insensitivity to local deformation, we explore the statistical properties of the second-order NAST of a Gaussian process with various dependence structures and its interaction with the chosen wavelets and activation functions. We also provide central limit theorem (CLT) and non-CLT results. Numerical simulations demonstrate the developed theorems. Our results explain how NAST processes complicated time series, paving a way toward statistical inference based on NAST for real-world applications.A central limit theorem for the mean starting hitting time for a random walk on a random graphhttps://zbmath.org/1516.600172023-09-19T14:22:37.575876Z"Löwe, Matthias"https://zbmath.org/authors/?q=ai:lowe.matthias"Terveer, Sara"https://zbmath.org/authors/?q=ai:terveer.sara-karinaThe authors establish a central limit theorem for hitting times in a random walk on an Erdős--Rényi random graph. Consider the graph \(G(n,p_n)\) with \(n\) vertices and each pair of vertices joined by an edge with probability \(p_n\) such that \(np_n^2/(\log n)^{16\xi}\to\infty\) as \(n\to\infty\), where \(\xi>1\) is a sufficiently large constant independent of \(n\). This choice ensures that the graph is asymptotically almost surely connected. To construct a sequence of such graphs, the authors increment the indicators of the existence of an edge between vertices \(i\) and \(j\) via an appropriate time-inhomogeneous Markov chain for each \(\{i,j\}\) with \(i\not=j\). On a given graph with fixed \(n\), a discrete-time random walk evolves by choosing a neighbour of the current location to which to move, each with equal probability. Let \(H_{i,j}\) denote the expected time such a random walk takes to reach vertex \(j\), when started from vertex \(i\). The main result of the present paper is a central limit theorem for \(H=\sum_j\pi_jH_{i,j}\) for fixed \(i\), where \(\pi_j\) is the stationary probability at vertex \(j\) for the random walk, which is proportional to the degree of vertex \(j\). That is, the authors show that, suitably normalized, \(H\) converges in distribution to Gaussian as \(n\to\infty\). The normalization and parameters of the limiting Gaussian distribution are given explicitly.
Reviewer: Fraser Daly (Edinburgh)Large fluctuations and transport properties of the Lévy-Lorentz gashttps://zbmath.org/1516.600182023-09-19T14:22:37.575876Z"Zamparo, Marco"https://zbmath.org/authors/?q=ai:zamparo.marcoSummary: The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a precise large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that the asymptotic diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion, and not superdiffusion. Although the Lévy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings suggest that superdiffusion is a transient behavior which develops into normal diffusion on long timescales, and raise a new question about how the transition from the quenched normal diffusion to the annealed superdiffusion occurs.The harmonic mean formula for random processeshttps://zbmath.org/1516.600202023-09-19T14:22:37.575876Z"Bisewski, Krzysztof"https://zbmath.org/authors/?q=ai:bisewski.krzysztof"Hashorva, Enkelejd"https://zbmath.org/authors/?q=ai:hashorva.enkelejd"Shevchenko, Georgiy"https://zbmath.org/authors/?q=ai:shevchenko.georgiy-mSummary: Motivated by the classical harmonic mean formula, estabished by
\textit{D. Aldous} in [Discrete Math. 76, No. 3, 167--176 (1989; Zbl 0681.05062)], we investigate the relation between the sojourn time and supremum of a random process \(X(t)\), \(t \in \mathbb{R}^d\) and extend the harmonic mean formula for general stochastically continuous \(X\). We discuss two applications concerning the continuity of distribution of supremum of \(X\) and representations of classical Pickands constants.Fractional evolution equation with Cauchy data in \(L^p\) spaceshttps://zbmath.org/1516.600212023-09-19T14:22:37.575876Z"Phuong, Nguyen Duc"https://zbmath.org/authors/?q=ai:phuong.nguyen-duc"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinhSummary: In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in \(L^2\) and \(H^s\). However, there have not been any papers dealing with this problem with observed data in \(L^p\) with \(p\neq 2\). We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in \(L^p\). To our knowledge, \(L^p\) evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.Linear multifractional stable sheets in the broad sense: existence and joint continuity of local timeshttps://zbmath.org/1516.600222023-09-19T14:22:37.575876Z"Ding, Yujia"https://zbmath.org/authors/?q=ai:ding.yujia"Peng, Qidi"https://zbmath.org/authors/?q=ai:peng.qidi"Xiao, Yimin"https://zbmath.org/authors/?q=ai:xiao.yiminSummary: We introduce the notion of \textit{linear multifractional stable sheets in the broad sense} (LMSS) with \(\alpha \in (0,2]\), to include both linear multifractional Brownian sheets (\( \alpha =2\)) and linear multifractional stable sheets (\( \alpha < 2\)). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a \textit{sufficient and necessary condition} for the existence of local times of LMSS; Theorem 3.1 shows a \textit{sufficient condition} for the joint continuity of local times; and Theorem 4.1 proves a sharp local Hölder condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.On the fractional stochastic integration for random non-smooth integrandshttps://zbmath.org/1516.600232023-09-19T14:22:37.575876Z"Dokuchaev, Nikolai"https://zbmath.org/authors/?q=ai:dokuchaev.nikolai-gSummary: The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter \(H > 1 /2\). The integral is defined initially on the processes that are`` piecewise'' predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations. On the other hand, a pathwise regularity, such as Hölder condition, etc., is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring certain restrictions on the forecasting error. This integration is based on Itô's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend right continuously on \(H\) at \(H = 1/2\).An adjoint-free four-dimensional variational data assimilation method via a modified Cholesky decomposition and an iterative Woodbury matrix formulahttps://zbmath.org/1516.600242023-09-19T14:22:37.575876Z"Nino-Ruiz, Elias D."https://zbmath.org/authors/?q=ai:nino-ruiz.elias-david"Guzman-Reyes, Luis G."https://zbmath.org/authors/?q=ai:guzman-reyes.luis-g"Beltran-Arrieta, Rolando"https://zbmath.org/authors/?q=ai:beltran-arrieta.rolando(no abstract)With or without replacement? Sampling uncertainty in Shepp's urn schemehttps://zbmath.org/1516.600252023-09-19T14:22:37.575876Z"Glover, Kristoffer"https://zbmath.org/authors/?q=ai:glover.kristoffer-jSummary: We introduce a variant of Shepp's classical urn problem in which the optimal stopper does not know whether sampling from the urn is done with or without replacement. By considering the problem's continuous-time analog, we provide bounds on the value function and, in the case of a balanced urn (with an equal number of each ball type), an explicit solution is found. Surprisingly, the optimal strategy for the balanced urn is the same as in the classical urn problem. However, the expected value upon stopping is lower due to the additional uncertainty present.Convergence of uniformly Pettis integrable multivalued martingaleshttps://zbmath.org/1516.600262023-09-19T14:22:37.575876Z"El-louh, M'hamed"https://zbmath.org/authors/?q=ai:el-louh.mhamed"El Allali, Mohammed"https://zbmath.org/authors/?q=ai:el-allali.mohammed"Ezzaki, Fatima"https://zbmath.org/authors/?q=ai:ezzaki.fatimaSummary: Some various convergence results are established for both uniformly Pettis integrable martingales with convex weakly compact values and Pettis integrable martingales with convex closed and bounded values in a separable Banach space. A Castaing representation of its Pettis integrable multivalued martingales is proved. Some properties of Pettis integrable multivalued functions are provided.Cutoff for rewiring dynamics on perfect matchingshttps://zbmath.org/1516.600272023-09-19T14:22:37.575876Z"Olesker-Taylor, Sam"https://zbmath.org/authors/?q=ai:olesker-taylor.samSummary: We establish cutoff for a natural random walk \((\mathsf{RW})\) on the set of perfect matchings \((\mathsf{PM}\)s), based on ``rewiring''. An \(n\)-\(\mathsf{PM}\) is a pairing of \(2n\) objects. The \(k\)-\(\mathsf{PM}\;\mathsf{RW}\) selects \(k\) pairs uniformly at random, disassociates the corresponding \(2k\) objects, then chooses a new pairing on these \(2k\) objects uniformly at random. The equilibrium distribution is uniform over all \(n\)-\(\mathsf{PM}\)s.
The \(2\)-\(\mathsf{PM}\;\mathsf{RW}\) was first introduced by \textit{P. W. Diaconis} and \textit{S. P. Holmes} [Proc. Natl. Acad. Sci. USA 95, No. 25, 14600--14602 (1998; Zbl 0908.92023)], seen as a \(\mathsf{RW}\) on phylogenetic trees. They established cutoff in this case. We establish cutoff for the \(k\)-\(\mathsf{PM}\;\mathsf{RW}\) whenever \(2\leq k\ll n\). If \(k\gg 1\), then the mixing time is \(\frac{n}{k}\log n\) to leading order.
\textit{P. Diaconis} and \textit{S. Holmes} [Electron. J. Probab. 7, Paper No. 6, 17 p. (2002; Zbl 1007.60071)] relate the \(2\)-\(\mathsf{PM}\; \mathsf{RW}\) to the random transpositions card shuffle. \textit{T. Ceccherini-Silberstein} et al. [J. Math. Sci., New York 141, No. 2, 1182--1229 (2007; Zbl 1173.43001); translation from Sovrem. Mat. Prilozh. 27, 95--140 (2005); Harmonic analysis on finite groups. Representation theory, Gelfand pairs and Markov chains. Cambridge: Cambridge University Press (2008; Zbl 1149.43001)] establish the same result using representation theory. We are the first to handle \(k>2\). We relate the \(\mathsf{PM}\;\mathsf{RW}\) to conjugacy-invariant \(\mathsf{RW}\)s on the permutation group by introducing a ``cycle structure'' for \(\mathsf{PM}\)s, then build on work of \textit{N. Berestycki} et al. [Ann. Probab. 39, No. 5, 1815--1843 (2011; Zbl 1245.60006)], \textit{N. Berestycki} and \textit{B. Şengül} [Probab. Theory Relat. Fields 173, No. 3--4, 1197--1241 (2019; Zbl 1411.60012)] and \textit{O. Schramm} [Isr. J. Math. 147, 221--243 (2005; Zbl 1130.60302)] on such \(\mathsf{RW}\)s.On moments of downward passage times for spectrally negative Lévy processeshttps://zbmath.org/1516.600282023-09-19T14:22:37.575876Z"Behme, Anita"https://zbmath.org/authors/?q=ai:behme.anita-diana"Strietzel, Philipp Lukas"https://zbmath.org/authors/?q=ai:strietzel.philipp-lukasSummary: The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to \(+\infty, -\infty\), or oscillating. Whenever the Lévy process drifts to \(+\infty\), we prove that the \(\kappa\)th moment of the first passage time (conditioned to be finite) exists if and only if the \((\kappa+1)\)th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér-Lundberg risk processes. Whenever the Lévy process drifts to \(-\infty\), we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.A decomposition for Lévy processes inspected at Poisson momentshttps://zbmath.org/1516.600292023-09-19T14:22:37.575876Z"Boxma, Onno"https://zbmath.org/authors/?q=ai:boxma.onno-j"Mandjes, Michel"https://zbmath.org/authors/?q=ai:mandjes.michelSummary: We consider a Lévy process \(Y(t)\) that is not continuously observed, but rather inspected at Poisson\((\omega)\) moments only, over an exponentially distributed time \(T_\beta\) with parameter \(\beta\). The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to \(T_\beta\), denoted by \(Y_{\beta, \omega}\). Our main result is a decomposition: we derive a remarkable distributional equality that contains \(Y_{\beta, \omega}\) as well as the running maximum process \(\bar{Y}(t)\) at the exponentially distributed times \(T_\beta\) and \(T_{\beta+\omega}\). Concretely, \(\overline{Y}(T_\beta)\) can be written as the sum of two independent random variables that are distributed as \(Y_{\beta, \omega}\) and \(\overline{Y}(T_{\beta+\omega})\). The distribution of \(Y_{\beta, \omega}\) can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér-Lundberg insurance risk model.Gap probability for the hard edge Pearcey processhttps://zbmath.org/1516.600302023-09-19T14:22:37.575876Z"Dai, Dan"https://zbmath.org/authors/?q=ai:dai.dan"Xu, Shuai-Xia"https://zbmath.org/authors/?q=ai:xu.shuaixia"Zhang, Lun"https://zbmath.org/authors/?q=ai:zhang.lunSummary: The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. This paper deals with the gap probability for the thinned/unthinned hard edge Pearcey process over the interval \((0, s)\) by working on a \(3 \times 3\) matrix-valued Riemann-Hilbert problem for the relevant Fredholm determinants. We establish an integral representation of the gap probability via the Hamiltonian related to a new system of coupled differential equations. Together with some remarkable differential identities for the Hamiltonian, we derive the large gap asymptotics for the thinned hard edge Pearcey process, including the explicitly evaluation of the constant factor in terms of the Barnes G-function. As an application, we also obtain the asymptotic statistical properties of the counting function for the hard edge Pearcey process.On the cumulant transforms for Hawkes processeshttps://zbmath.org/1516.600312023-09-19T14:22:37.575876Z"Lee, Young"https://zbmath.org/authors/?q=ai:lee.young-duck|lee.young-jack|lee.young-su|lee.young-whan|lee.young-im|lee.young-shiuan|lee.young-sam|lee.young-gill|lee.young-yoon|lee.young-ran|lee.young-kow|lee.young-he|lee.young-kyun|lee.young-chan|lee.young-seop|lee.young-seo|lee.young-hae|lee.young-joo|lee.young-seek|lee.young-kyu|lee.young-sun|lee.young-ram|lee.young-shin|lee.younglok|lee.young-choon|lee.young-jin|lee.young-doo|lee.young-ok|lee.young-moo|lee.young-soo|lee.young-cheon|lee.young-sup|lee.young-han|lee.young-seol|lee.young-chen|lee.young-hoon|lee.young-jae|lee.young-ro"Rheinländer, Thorsten"https://zbmath.org/authors/?q=ai:rheinlander.thorstenSummary: We consider the asset price as the weak solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale whose predictable compensator follows shot-noise and Hawkes processes. In this framework, we discuss the Esscher martingale measure where the conditions for its existence are detailed. This generalizes certain relationships not yet encountered in the literature.On the splitting and aggregating of Hawkes processeshttps://zbmath.org/1516.600322023-09-19T14:22:37.575876Z"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.10"Pang, Guodong"https://zbmath.org/authors/?q=ai:pang.guodongSummary: We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration-birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).Small scale CLTs for the nodal length of monochromatic waveshttps://zbmath.org/1516.600332023-09-19T14:22:37.575876Z"Dierickx, Gauthier"https://zbmath.org/authors/?q=ai:dierickx.gauthier"Nourdin, Ivan"https://zbmath.org/authors/?q=ai:nourdin.ivan"Peccati, Giovanni"https://zbmath.org/authors/?q=ai:peccati.giovanni"Rossi, Maurizia"https://zbmath.org/authors/?q=ai:rossi.mauriziaSummary: We consider the nodal length \(L(\lambda)\) of the restriction to a ball of radius \(r_\lambda\) of a \textit{Gaussian pullback monochromatic random wave} of parameter \(\lambda >0\) associated with a Riemann surface \((\mathcal M,g)\) without conjugate points. Our main result is that, if \(r_\lambda\) grows slower than \((\log \lambda)^{1/25}\), then (as \(\lambda \rightarrow \infty)\) the length \(L(\lambda)\) verifies a Central Limit Theorem with the same scaling as Berry's random wave model -- as established in [\textit{I. Nourdin} et al., Commun. Math. Phys. 369, No. 1, 99--151 (2019; Zbl 1431.60025)]. Taking advantage of some powerful extensions of an estimate by \textit{P. H. Bérard} [Math. Z. 155, 249--276 (1977; Zbl 0341.35052)] due to \textit{B. Keeler} [``A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points'', Preprint, \url{arXiv:1905.05136}], our techniques are mainly based on a novel intrinsic bound on the coupling of smooth Gaussian fields, that is of independent interest, and moreover allow us to improve some estimates for the nodal length asymptotic variance of pullback random waves in [\textit{Y. Canzani} and \textit{B. Hanin}, Commun. Math. Phys. 378, No. 3, 1677--1712 (2020; Zbl 1476.58031)]. In order to demonstrate the flexibility of our approach, we also provide an application to phase transitions for the nodal length of arithmetic random waves on shrinking balls of the 2-torus.Hitting times, number of jumps, and occupation times for continuous-time finite state Markov chainshttps://zbmath.org/1516.600442023-09-19T14:22:37.575876Z"Colwell, David B."https://zbmath.org/authors/?q=ai:colwell.david-bSummary: In this paper we discuss three random variables associated with a finite state continuous-time Markov chain having a constant transition rate matrix: the number of jumps between two different states, the amount of time spent in a given state, known as an occupation time, and the time it takes for the Markov chain to first reach a particular state, known as a hitting time or first passage time. First, we calculate expected values for each of these random variables. We then derive the moment generating functions for these variables. Our approach is to use the vector/matrix representations of the various processes. Each solution is written in terms of an integral of the exponential of a matrix.Wasserstein perturbations of Markovian transition semigroupshttps://zbmath.org/1516.600452023-09-19T14:22:37.575876Z"Fuhrmann, Sven"https://zbmath.org/authors/?q=ai:fuhrmann.sven"Kupper, Michael"https://zbmath.org/authors/?q=ai:kupper.michael"Nendel, Max"https://zbmath.org/authors/?q=ai:nendel.maxSummary: In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modelled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of Lévy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.Equivalence of Liouville measure and Gaussian free fieldhttps://zbmath.org/1516.600492023-09-19T14:22:37.575876Z"Berestycki, Nathanaël"https://zbmath.org/authors/?q=ai:berestycki.nathanael"Sheffield, Scott"https://zbmath.org/authors/?q=ai:sheffield.scott"Sun, Xin"https://zbmath.org/authors/?q=ai:sun.xinSummary: Given an instance \(h\) of the Gaussian free field on a planar domain \(D\) and a constant \(\gamma \in (0,2)\), one can use various regularization procedures to make sense of the \textit{Liouville quantum gravity area measure} \(\mu :={e^{\gamma h(z)}}dz\). It is known that the field \(h\) a.s. determines the measure \({\mu_h}\). We show that the converse is true: namely, \(h\) is measurably determined by \({\mu_h}\). More generally, given a random closed fractal subset \(\mathcal{A}\) endowed with a Frostman measure \(\sigma\) whose support is \(\mathcal{A}\) (independent of \(h)\), a Gaussian multiplicative chaos measure \({\mu_{\sigma ,h}}\) can be constructed. We give a mild condition on \((\mathcal{A},\sigma)\) under which \({\mu_{\sigma ,h}}\) determines \(h\) restricted to \(\mathcal{A}\), in the sense that it determines its harmonic extension off \(\mathcal{A}\). Our condition is satisfied by the occupation measures of planar Brownian motion and SLE curves under natural parametrizations. Along the way we obtain general positive moment bounds for Gaussian multiplicative chaos. Contrary to previous results, this does not require any assumption on the underlying measure \(\sigma\) such as scale invariance, and hence may be of independent interest.Truncated Lévy flights and generalized Cauchy processeshttps://zbmath.org/1516.600512023-09-19T14:22:37.575876Z"Lubashevsky, I. A."https://zbmath.org/authors/?q=ai:lubashevskii.i-aSummary: A continuous Markovian model for truncated Lévy flights is proposed. It generalizes the approach developed previously by \textit{I. Lubashevsky} et al.
[Phys. Rev. E (3) 79, No. 1, Article ID 011110, 5 p. (2009; \url{doi:10.1103/PhysRevE.79.011110});
Phys. Rev. E (3) 80, No. 3, Article ID 031148, 14 p. (2009; \url{doi:10.1103/PhysRevE.80.031148});
Eur. Phys. J. B, Condens. Matter Complex Syst. 78, No. 2, 207--216 (2010; Zbl 1256.60026)]
and allows for nonlinear friction in wandering particle motion as well as saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and, as shown in the paper, individually give rise to a cutoff in the generated random walks meeting the Lévy type statistics on intermediate scales. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method. The obtained numerical data were employed to analyze the statistics of the particle displacement during a given time interval, namely, to calculate the geometric mean of this random variable and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Lévy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Lévy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather than their characteristics individually.Extinction times of multitype continuous-state branching processeshttps://zbmath.org/1516.600522023-09-19T14:22:37.575876Z"Chaumont, Loïc"https://zbmath.org/authors/?q=ai:chaumont.loic"Marolleau, Marine"https://zbmath.org/authors/?q=ai:marolleau.marineSummary: A multitype continuous-state branching process (MCSBP) \(\mathrm{Z}={({\mathrm{Z}_t})_{t\ge 0}}\), is a Markov process with values in \({[0,\infty)^d}\) that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of \(d\) Laplace exponents of \({\mathbb{R}^d}\)-valued spectrally positive Lévy processes, each one having \(d-1\) increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for \(d=1\). Our arguments bear on elements of fluctuation theory for spectrally positive additive Lévy fields recently obtained in [\textit{L. Chaumont} and \textit{M. Marolleau}, Electron. J. Probab. 25, Paper No. 161, 26 p. (2020; Zbl 1477.60072)] and an extension of the Lamperti representation in higher dimension proved in [\textit{M. E. Caballero} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 3, 1280--1304 (2017; Zbl 1378.60111)].Reflection principle for finite-velocity random motionshttps://zbmath.org/1516.600582023-09-19T14:22:37.575876Z"Cinque, Fabrizio"https://zbmath.org/authors/?q=ai:cinque.fabrizioSummary: We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.Variational Tobit Gaussian process regressionhttps://zbmath.org/1516.620042023-09-19T14:22:37.575876Z"Basson, Marno"https://zbmath.org/authors/?q=ai:basson.marno"Louw, Tobias M."https://zbmath.org/authors/?q=ai:louw.tobias-m"Smith, Theresa R."https://zbmath.org/authors/?q=ai:smith.theresa-rSummary: We propose a variational inference-based framework for training a Gaussian process regression model subject to censored observational data. Data censoring is a typical problem encountered during the data gathering procedure and requires specialized techniques to perform inference since the resulting probabilistic models are typically analytically intractable. In this article we exploit the variational sparse Gaussian process inducing variable framework and local variational methods to compute an analytically tractable lower bound on the true log marginal likelihood of the probabilistic model which can be used to perform Bayesian model training and inference. We demonstrate the proposed framework on synthetically-produced, noise-corrupted observational data, as well as on a real-world data set, subject to artificial censoring. The resulting predictions are comparable to existing methods to account for data censoring, but provides a significant reduction in computational cost.Scalable computations for nonstationary Gaussian processeshttps://zbmath.org/1516.620052023-09-19T14:22:37.575876Z"Beckman, Paul G."https://zbmath.org/authors/?q=ai:beckman.paul-g"Geoga, Christopher J."https://zbmath.org/authors/?q=ai:geoga.christopher-j"Stein, Michael L."https://zbmath.org/authors/?q=ai:stein.michael-l"Anitescu, Mihai"https://zbmath.org/authors/?q=ai:anitescu.mihaiSummary: Nonstationary Gaussian process models can capture complex spatially varying dependence structures in spatial data. However, the large number of observations in modern datasets makes fitting such models computationally intractable with conventional dense linear algebra. In addition, derivative-free or even first-order optimization methods can be slow to converge when estimating many spatially varying parameters. We present here a computational framework that couples an algebraic block-diagonal plus low-rank covariance matrix approximation with stochastic trace estimation to facilitate the efficient use of second-order solvers for maximum likelihood estimation of Gaussian process models with many parameters. We demonstrate the effectiveness of these methods by simultaneously fitting 192 parameters in the popular nonstationary model of Paciorek and Schervish using 107,600 sea surface temperature anomaly measurements.Fitting Matérn smoothness parameters using automatic differentiationhttps://zbmath.org/1516.620142023-09-19T14:22:37.575876Z"Geoga, Christopher J."https://zbmath.org/authors/?q=ai:geoga.christopher-j"Marin, Oana"https://zbmath.org/authors/?q=ai:marin.oana"Schanen, Michel"https://zbmath.org/authors/?q=ai:schanen.michel"Stein, Michael L."https://zbmath.org/authors/?q=ai:stein.michael-lSummary: The Matérn covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter \(\nu\) gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Matérn covariance function with respect to \(\nu\) require derivatives of the modified second-kind Bessel function \(\mathcal{K}_\nu\) with respect to \(\nu\). While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing \(\nu\) as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating \(\nu\) use finite difference estimates for \(\partial_\nu\mathcal{K}_\nu\). In this work, we introduce a new implementation of \(\mathcal{K}_\nu\) that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. We provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.Robust numerical methods for nonlocal (and local) equations of porous medium type. II: Schemes and experimentshttps://zbmath.org/1516.650662023-09-19T14:22:37.575876Z"del Teso, Félix"https://zbmath.org/authors/?q=ai:del-teso.felix"Endal, Jørgen"https://zbmath.org/authors/?q=ai:endal.jorgen"Jakobsen, Espen R."https://zbmath.org/authors/?q=ai:jakobsen.espen-robstadSummary: We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations \(\partial_tu-\mathfrak L[\varphi(u)]=f(x,t)\) in \(\mathbb R^N\times(0,T)\), where \(\mathfrak L\) is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., \(\mathfrak L=\Delta\) or \(\mathfrak L=-(-\Delta)^{\frac{\alpha}{2}}\), \(\alpha\in(0,2)\), and porous medium, fast diffusion, and Stefan-type nonlinearities \(\phi\). By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are \(L^p\)-stable for \(p\in[1,\infty]\), compact, and convergent in \(C([0,T];L_{\mathrm{loc}}^p(\mathbb R^N))\) for \(p\in[1,\infty)\). The first part of this project is given in [\url{arXiv:1801.07148}] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I [the authors, SIAM J. Numer. Anal. 57, No. 5, 2266--2299 (2019; Zbl 1428.65005)] apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.Bayesian calibration of a numerical code for prediction. Theory of code calibration and application to the prediction of a photovoltaic power plant electricity productionhttps://zbmath.org/1516.651102023-09-19T14:22:37.575876Z"Carmassi, Mathieu"https://zbmath.org/authors/?q=ai:carmassi.mathieu"Barbillon, Pierre"https://zbmath.org/authors/?q=ai:barbillon.pierre"Chiodetti, Matthieu"https://zbmath.org/authors/?q=ai:chiodetti.matthieu"Keller, Merlin"https://zbmath.org/authors/?q=ai:keller.merlin"Parent, Éric"https://zbmath.org/authors/?q=ai:parent.ericSummary: Field experiments are often difficult and expensive to carry out. To bypass these issues, industrial companies have developed computational codes. These codes are intended to be representative of the physical system, but come with a certain number of problems. Despite continuous code development, the difference between the code outputs and experiments can remain significant. Two kinds of uncertainties are observed. The first one comes from the difference between the physical phenomenon and the values recorded experimentally. The second concerns the gap between the code and the physical system. To reduce this difference, often named model bias, discrepancy, or model error, computer codes are generally complexified in order to make them more realistic. These improvements increase the computational cost of the code. Moreover, a code often depends on user-defined parameters in order to match field data as closely as possible. This estimation task is called calibration. This paper proposes a review of Bayesian calibration methods and is based on an application case which makes it possible to discuss the various methodological choices and to illustrate their divergences. This example is based on a code used to predict the power of a photovoltaic plant.Differential quadrature method for space-fractional diffusion equations on 2D irregular domainshttps://zbmath.org/1516.651132023-09-19T14:22:37.575876Z"Zhu, X. G."https://zbmath.org/authors/?q=ai:zhu.xiaogang"Yuan, Z. B."https://zbmath.org/authors/?q=ai:yuan.zhanbin"Liu, F."https://zbmath.org/authors/?q=ai:liu.fawang"Nie, Y. F."https://zbmath.org/authors/?q=ai:nie.yufengSummary: In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by Lévy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.Do(es the influence of) empty waves survive in configuration space?https://zbmath.org/1516.810102023-09-19T14:22:37.575876Z"Durt, T."https://zbmath.org/authors/?q=ai:durt.thomasSummary: The de Broglie-Bohm interpretation is a no-collapse interpretation, which implies that we are in principle surrounded by empty waves generated by all particles of the universe, empty waves that will never collapse. It is common to establish an analogy between these pilot-waves and 3D radio-waves, which are nearly devoided of energy but carry nevertheless information to which we may have access after an amplification process. Here we show that this analogy is limited: if we consider empty waves in configuration space, an effective collapse occurs when a detector clicks and the 3ND empty wave associated to a particle may not influence another particle (even if these two particles are identical, e.g. bosons as in the example considered here).An investigation of continuous-time quantum walk on hypercube in view of Cartesian product structurehttps://zbmath.org/1516.810932023-09-19T14:22:37.575876Z"Han, Qi"https://zbmath.org/authors/?q=ai:han.qi.1"Kou, Yaxin"https://zbmath.org/authors/?q=ai:kou.yaxin"Wang, Huan"https://zbmath.org/authors/?q=ai:wang.huan.7"Bai, Ning"https://zbmath.org/authors/?q=ai:bai.ningSummary: In this paper, continuous-time quantum walk on hypercube is discussed in view of Cartesian product structure. We find that the \(n\)-fold Cartesian power of the complete graph \(K_2\) is the \(n\)-dimensional hypercube, which give us new ideas for the study of quantum walk on hypercube. Combining the product structure, the spectral distribution of the graph and the quantum decomposition of the adjacency matrix, the probability amplitudes of the continuous-time quantum walker's position at time \(t\) are given, and it is discussed that the probability distribution for the continuous-time case is uniform when \(t=(\pi\slash4)n\). The application of this product structure greatly improves the study of quantum walk on complex graphs, which has far-reaching influence and great significance.Spatial search algorithms on graphs with multiple targets using discrete-time quantum walkhttps://zbmath.org/1516.810942023-09-19T14:22:37.575876Z"Xue, Xi-Ling"https://zbmath.org/authors/?q=ai:xue.xiling"Sun, Zhi-Hong"https://zbmath.org/authors/?q=ai:sun.zhihong"Ruan, Yue"https://zbmath.org/authors/?q=ai:ruan.yue"Li, Xue"https://zbmath.org/authors/?q=ai:li.xueSummary: Search algorithms based on discrete-time quantum walk (QW) can be considered as alterations of the standard QW: Use a different coin operator that distinguishes target and nontarget vertices, or, mark the target vertices first followed by the standard QW. Two most frequently used marking coins are \(-I\) and \(D\) the negative identity operator and the negative Grover diffusion operator. We show that search algorithms corresponding to these four combinations can be reduced to two, denoted as \(U_I^\prime\) and \(U_D^\prime \), and they are equivalent when searching for nonadjacent multiple targets. For adjacent target vertices, numerical simulations show that the performance of the algorithm \(U_D^\prime\) highly depends on the density of the underlying graph, and it outperforms \(U_I^\prime\) when the density is large enough. At last, a generalized stationary state of both search algorithms on the graphs with even-numbered degree is provided.The Kossakowski matrix and strict positivity of Markovian quantum dynamicshttps://zbmath.org/1516.811142023-09-19T14:22:37.575876Z"Agredo, Julián"https://zbmath.org/authors/?q=ai:agredo.julian"Fagnola, Franco"https://zbmath.org/authors/?q=ai:fagnola.franco"Poletti, Damiano"https://zbmath.org/authors/?q=ai:poletti.damianoSummary: We investigate the relationship between strict positivity of the Kossakowski matrix, irreducibility and positivity improvement properties of Markovian quantum dynamics. We show that for a Gaussian quantum dynamical semigroup strict positivity of the Kossakowski matrix implies irreducibility and, with an additional technical assumption, that the support of any initial state is the whole space for any positive time.Reaction diffusion systems and extensions of quantum stochastic processeshttps://zbmath.org/1516.811442023-09-19T14:22:37.575876Z"Greenman, Chris D."https://zbmath.org/authors/?q=ai:greenman.chris-dSummary: Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued Itô calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth-death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.Abstract model of continuous-time quantum walk based on Bernoulli functionals and perfect state transferhttps://zbmath.org/1516.811452023-09-19T14:22:37.575876Z"Wang, Ce"https://zbmath.org/authors/?q=ai:wang.ceSummary: In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let \(\mathfrak{h}\) be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace \(\mathfrak{h}_L\subset\mathfrak{h}\) a self-adjoint operator \(\Delta_L\) via the canonical unitary involutions on \(\mathfrak{h} \), and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with \(\mathfrak{h}_L\) as its state space, which is governed by the Schrödinger equation with \(\Delta_L\) as the Hamiltonian. We define the time-average probability distribution of the model, obtain an explicit expression of the distribution, and, especially, we find the distribution admits a symmetry property. We also justify the model by offering a graph-theoretic interpretation to the operator \(\Delta_L\) as well as to the model itself. Finally, we prove that the model has PST at time \(t=\frac{ \pi}{ 2} \). Some other interesting results about the model are also proved.Mass distributions of dijet resonances from excited quarks at proton-proton collidershttps://zbmath.org/1516.811792023-09-19T14:22:37.575876Z"Guler, Emine Gurpinar"https://zbmath.org/authors/?q=ai:guler.emine-gurpinar"Guler, Yalcin"https://zbmath.org/authors/?q=ai:guler.yalcin"Harris, Robert M."https://zbmath.org/authors/?q=ai:harris.robert-michaelSummary: We study the expected experimental mass distributions of dijet resonances from excited quarks in proton-proton collisions at energies \(\sqrt{s} = 13\), 14, 27, 100, 300, and 500 TeV. We explore in detail the expected shapes at both the generator and experimental levels, and identify within the distributions the effects of the excited quark natural width, parton momentum distributions of the proton, radiation, and experimental resolution. We present both differential and cumulative probability distributions as a function of dijet mass, and the signal acceptance of a window in dijet mass centered on each resonance. We find that for a range of resonance masses, between 10\% and 50\% of \(\sqrt{s}\), the dijet mass distributions and window acceptance are practically universal, approximately invariant under changes in resonance mass and \(\sqrt{s}\). This universality is violated when the resonance mass reaches 60\% of \(\sqrt{s}\), because the steepness of the parton momentum distributions of the proton produces a significant tail at low dijet mass. This work supports our Snowmass 2021 study [``Sensitivity to dijet resonances at proton-proton colliders'', Preprint, \url{arXiv:2202.03389}]on the sensitivity to dijet resonances at proton-proton colliders.Single production of vectorlike \(B\) quarks decaying into \(bh\) at the CLIChttps://zbmath.org/1516.811812023-09-19T14:22:37.575876Z"Han, Jin-Zhong"https://zbmath.org/authors/?q=ai:han.jinzhong"Xu, Shuai"https://zbmath.org/authors/?q=ai:xu.shuai"Song, Hong-Quan"https://zbmath.org/authors/?q=ai:song.hong-quan"Wang, Yu-Jie"https://zbmath.org/authors/?q=ai:wang.yu-jieSummary: New vectorlike quarks have been predicted in many scenarios of new physics beyond the Standard Model, which may be potentially discovered at the current and future high-energy colliders. Based on a model-independent framework, we study the single production process of the doublet vectorlike \(B\)-quark (VLQ-\(B\)) at the future Compact Linear Collider (CLIC) and explore the possible signals through the decay channel \(B \to bh\) with subsequent decay mode of the Higgs boson: \(h\to WW^\ast\to\ell\nu jj\). By performing a rapid detector simulation of the signal and background events as well as considering the initial state radiation and beamstrahlung effects, the exclusion limit at the 95\% confidence level and the \(5\sigma\) discovery reach in the parameter spaces (the coupling strength \(\kappa_B\) and the VLQ-\(B\) mass) are obtained at the future 3 TeV CLIC with an integrated luminosity of 5 \(\mathrm{ab}^{-1}\).Di-photon decay of a light Higgs state in the BLSSMhttps://zbmath.org/1516.812222023-09-19T14:22:37.575876Z"Abdelalim, Ahmed Ali"https://zbmath.org/authors/?q=ai:abdelalim.ahmed-ali"Das, Biswaranjan"https://zbmath.org/authors/?q=ai:das.biswaranjan"Khalil, Shaaban"https://zbmath.org/authors/?q=ai:khalil.shaaban"Moretti, Stefano"https://zbmath.org/authors/?q=ai:moretti.stefano.1Summary: In the context of the \(B - L\) Supersymmetric Standard Model (BLSSM), we investigate the consistency of a scenario with a light Higgs boson with mass in the range 94--98 GeV and a Standard Model (SM) Higgs state at 125 GeV with the results of a search performed by the CMS Collaboration in the di-photon channel primarily involving data at an integrated luminosity of 35.9 \(\mathrm{fb}^{-1}\) and an energy of \(\sqrt{s} = 13\) TeV. In this study, we present a Monte Carlo (MC) analysis of signal and background mimicking the experimental one and showing acceptable consistency with data, at both the integral and differential level.Roughening transition and universality of single step growth models in (2+1)-dimensionshttps://zbmath.org/1516.821092023-09-19T14:22:37.575876Z"Dashti-Naserabadi, H."https://zbmath.org/authors/?q=ai:dashti-naserabadi.h"Saberi, A. A."https://zbmath.org/authors/?q=ai:saberi.abbas-ali"Rouhani, S."https://zbmath.org/authors/?q=ai:rouhani.shahinSummary: We study (2+1)-dimensional single step model for crystal growth including both deposition and evaporation processes parametrized by a single control parameter \(p\). Using extensive numerical simulations with a relatively high statistics, we estimate various interface exponents such as roughness, growth and dynamic exponents as well as various geometric and distribution exponents of height clusters and their boundaries (or iso-height lines) as function of \(p\). We find that, in contrary to the general belief, there exists a critical value \(p_c\approx 0.25\) at which the model undergoes a roughening transition from a rough phase with \(p<p_c\) in the Kardar-Parisi-Zhang universality to a smooth phase with \(p>p_c\), asymptotically in the Edwards-Wilkinson class. We validate our conclusion by estimating the effective roughness exponents and their extrapolation to the infinite-size limit.Electricity intraday price modelling with marked Hawkes processeshttps://zbmath.org/1516.910582023-09-19T14:22:37.575876Z"Deschatre, Thomas"https://zbmath.org/authors/?q=ai:deschatre.thomas"Gruet, Pierre"https://zbmath.org/authors/?q=ai:gruet.pierreThe authors consider a two-dimensional marked Hawkes process with increasing baseline intensity to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump sizes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modelled on those particular markets. The authors provide analytic formulas for first and second moments and for the signature plot in the context of Hawkes processes with random jump sizes and time-dependent baseline intensity. The model is estimated on German data.
Also, a result is provided about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.
Reviewer: Pavel Stoynov (Sofia)Diffusion approximations for periodically arriving expert opinions in a financial market with Gaussian drifthttps://zbmath.org/1516.910602023-09-19T14:22:37.575876Z"Sass, Jörn"https://zbmath.org/authors/?q=ai:sass.jorn"Westphal, Dorothee"https://zbmath.org/authors/?q=ai:westphal.dorothee"Wunderlich, Ralf"https://zbmath.org/authors/?q=ai:wunderlich.ralfThe authors study a financial market in which stock returns depend on an unobservable Gaussian drift process. Investors obtain information on that drift from return observations and discrete-time expert opinions as an external source of information. Estimates of the hidden drift process are based on filtering techniques in the case where the experts provide on the drift with high frequency and the variances grow linearly with the arrival frequency.
The asymptotic behavior of the filter as the arrival frequency tends to infinity is described by limit theorems. These state that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. \newline The authors apply the diffusion approximations of the filter for deriving simplified approximate solutions of utility maximization problems with logarithmic and power utility.
Reviewer: Pavel Stoynov (Sofia)Shadow price approximation for the fractional Black Scholes modelhttps://zbmath.org/1516.910612023-09-19T14:22:37.575876Z"Dolemweogo, Sibiri Narcisse"https://zbmath.org/authors/?q=ai:dolemweogo.sibiri-narcisse"Frédéric, Béré"https://zbmath.org/authors/?q=ai:frederic.bere"Clovis, Nitiéma Pierre"https://zbmath.org/authors/?q=ai:clovis.nitiema-pierreSummary: In this work, we used \textit{Nguyen Tien Dung} and \textit{Tran Hung Thao}'s approximation [Braz. J. Probab. Stat. 24, No. 1, 57--67 (2010; Zbl 1298.60060)] of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.\(N\)-fold compound option pricing with technical risk under fractional jump-diffusion modelhttps://zbmath.org/1516.910632023-09-19T14:22:37.575876Z"Zhao, Pingping"https://zbmath.org/authors/?q=ai:zhao.pingping"Xiang, Kaili"https://zbmath.org/authors/?q=ai:xiang.kaili"Chen, Peimin"https://zbmath.org/authors/?q=ai:chen.peiminSummary: The problem of generalizing the compound option pricing model to incorporate more empirical features becomes an urgent and necessary event. In this study, a new \(N\)-fold compound option pricing method is designed for the economic uncertainty and technical uncertainty. The economic uncertainty is modelled by a fractional jump-diffusion model, which incorporates the long-term dependence of financial markets, the kurtosis of returns and the unpredictable shocks of real world. The technical uncertainty is modelled as a simplified version Poisson-type jump process, which describes the catastrophic impact of the technical risk in multi-stage projects. The main contribution of this paper is that we firstly develop the \(N\)-fold compound option pricing model with the fractional Brownian motion and the technical risk variable. Further, the analytic solutions of pricing compound options are achieved and verified by the recursive formula of option price. Numerical examples are provided to support the theoretical results of this model. Moreover, from the sensitivity analysis, some results are presented to illustrate that the \(N\)-fold compound option price without considering the phase-specific characteristics of technical risk is improperly estimated. Thereby, it is essential for investors to take into account the technical risk when making decisions.Pricing high-dimensional Bermudan options with hierarchical tensor formatshttps://zbmath.org/1516.910662023-09-19T14:22:37.575876Z"Bayer, Christian"https://zbmath.org/authors/?q=ai:bayer.christian"Eigel, Martin"https://zbmath.org/authors/?q=ai:eigel.martin"Sallandt, Leon"https://zbmath.org/authors/?q=ai:sallandt.leon"Trunschke, Philipp"https://zbmath.org/authors/?q=ai:trunschke.philippSummary: An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality'' can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favorable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent neural network based methods.Donnelly (1983) and the limits of genetic genealogyhttps://zbmath.org/1516.920262023-09-19T14:22:37.575876Z"Edge, Michael D."https://zbmath.org/authors/?q=ai:edge.michael-d"Coop, Graham"https://zbmath.org/authors/?q=ai:coop.grahamConcerns [\textit{K. P. Donnelly}, ibid. 23, 34--63 (1983; Zbl 0521.92011)].Fast likelihood calculation for multivariate Gaussian phylogenetic models with shiftshttps://zbmath.org/1516.920582023-09-19T14:22:37.575876Z"Mitov, Venelin"https://zbmath.org/authors/?q=ai:mitov.venelin"Bartoszek, Krzysztof"https://zbmath.org/authors/?q=ai:bartoszek.krzysztof"Asimomitis, Georgios"https://zbmath.org/authors/?q=ai:asimomitis.georgios"Stadler, Tanja"https://zbmath.org/authors/?q=ai:stadler.tanjaSummary: Phylogenetic comparative methods (PCMs) have been used to study the evolution of quantitative traits in various groups of organisms, ranging from micro-organisms to animal and plant species. A common approach has been to assume a Gaussian phylogenetic model for the trait evolution along the tree, such as a branching Brownian motion (BM) or an Ornstein-Uhlenbeck (OU) process. Then, the parameters of the process have been inferred based on a given tree and trait data for the sampled species. At the heart of this inference lie multiple calculations of the model likelihood, that is, the probability density of the observed trait data, conditional on the model parameters and the tree. With the increasing availability of big phylogenetic trees, spanning hundreds to several thousand sampled species, this approach is facing a two-fold challenge. First, the assumption of a single Gaussian process governing the entire tree is not adequate in the presence of heterogeneous evolutionary forces acting in different parts of the tree. Second, big trees present a computational challenge, due to the time and memory complexity of the model likelihood calculation.
Here, we explore a sub-family, denoted \(\mathcal{G}_{\mathrm{L\,Inv}}\), of the Gaussian phylogenetic models, with the transition density exhibiting the properties that the expectation depends linearly on the ancestral trait value and the variance is invariant with respect to the ancestral value. We show that \(\mathcal{G}_{\mathrm{L\,Inv}}\) contains the vast majority of Gaussian models currently used in PCMs, while supporting an efficient (linear in the number of nodes) algorithm for the likelihood calculation. The algorithm supports scenarios with missing data, as well as different types of trees, including trees with polytomies and non-ultrametric trees. To account for the heterogeneity in the evolutionary forces, the algorithm supports models with ``shifts'' occurring at specific points in the tree. Such shifts can include changes in some or all parameters, as well as the type of the model, provided that the model remains within the \(\mathcal{G}_{\mathrm{L\,Inv}}\) family. This contrasts with most of the current implementations where, due to slow likelihood calculation, the shifts are restricted to specific parameters in a single type of model, such as the long-term selection optima of an OU process, assuming that all of its other parameters, such as evolutionary rate and selection strength, are global for the entire tree.
We provide an implementation of this likelihood calculation algorithm in an accompanying \(\mathtt{R}\)-package called PCMBase. The package has been designed as a generic library that can be integrated with existing or novel maximum likelihood or Bayesian inference tools.Host vector dynamics of a nonlinear pine wilt disease model in deterministic and stochastic environmentshttps://zbmath.org/1516.921182023-09-19T14:22:37.575876Z"Shi, Zhenfeng"https://zbmath.org/authors/?q=ai:shi.zhenfeng"Cao, Zhongwei"https://zbmath.org/authors/?q=ai:cao.zhongwei"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqingSummary: In this study, we develop a vector-host transmission model with general incidence rates for the dynamics of pine wilt disease in deterministic and stochastic environments. The existence and local asymptotic stability of equilibria are investigated in the deterministic case. We show the required conditions for the ergodic stationary distribution and extinction of the model in the stochastic case by constructing appropriate Lyapunov functions. Furthermore, by solving the corresponding Fokker-Planck equation, we obtain exact expressions of probability density function around the quasi-equilibrium of the stochastic model. Finally, we employ comprehensive numerical simulations to support our results and compare deterministic and stochastic situations.From the optimal singular stochastic control to the optimal stopping for regime-switching processeshttps://zbmath.org/1516.932762023-09-19T14:22:37.575876Z"Shao, Jinghai"https://zbmath.org/authors/?q=ai:shao.jinghai"Tian, Taoran"https://zbmath.org/authors/?q=ai:tian.taoranSummary: This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in
[\textit{U. G. Haussmann} and \textit{W. Suo}, SIAM J. Control Optim. 33, No. 3, 916--936 (1995; Zbl 0925.93958); ibid. 33, No. 3, 937--959 (1995; Zbl 0925.93959)].
In addition, our compactification method can remove the convexity condition on the coefficients used in
[\textit{F. Dufour} and \textit{B. Miller}, ibid. 43, No. 2, 708--730 (2004; Zbl 1101.93084)].