Recent zbMATH articles in MSC 60Ghttps://zbmath.org/atom/cc/60G2024-07-17T13:47:05.169476ZWerkzeugCovariance matrices of length power functionals of random geometric graphs -- an asymptotic analysishttps://zbmath.org/1536.054152024-07-17T13:47:05.169476Z"Reitzner, Matthias"https://zbmath.org/authors/?q=ai:reitzner.matthias"Römer, Tim"https://zbmath.org/authors/?q=ai:romer.tim"von Westenholz, Mandala"https://zbmath.org/authors/?q=ai:von-westenholz.mandalaSummary: Asymptotic properties of a vector of length power functionals of random geometric graphs are investigated. Algebraic properties of the asymptotic covariance matrix are studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a systematic discussion of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the formulation of the results a case distinction is necessary. In the three possible regimes the respective covariance matrix is of quite different nature which leads to different statements. Stochastic consequences for random geometric graphs are derived.The longest edge of the one-dimensional soft random geometric graph with boundarieshttps://zbmath.org/1536.054172024-07-17T13:47:05.169476Z"Rousselle, Arnaud"https://zbmath.org/authors/?q=ai:rousselle.arnaud"Sönmez, Ercan"https://zbmath.org/authors/?q=ai:sonmez.ercanSummary: The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant \(\alpha\), which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power \(\alpha\) and we recover a phase transition which coincides with exactly the same phases in \textit{I. Benjamini} and \textit{N. Berger} [Random Struct. Algorithms 19, No. 2, 102--111 (2001; Zbl 0983.60099)].Random walks on regular trees can not be slowed downhttps://zbmath.org/1536.054192024-07-17T13:47:05.169476Z"Angel, Omer"https://zbmath.org/authors/?q=ai:angel.omer"Richey, Jacob"https://zbmath.org/authors/?q=ai:richey.jacob"Spinka, Yinon"https://zbmath.org/authors/?q=ai:spinka.yinon"Yehudayoff, Amir"https://zbmath.org/authors/?q=ai:yehudayoff.amirSummary: We study a permuted random walk process on a graph \(G\). Given a fixed sequence of permutations on the vertices of \(G\), the permuted random walker alternates between taking random walk steps, and applying the next permutation in the sequence to their current position. Existing work on permuted random walks includes results on hitting times, mixing times, and asymptotic speed. The usual random walk on a regular tree, or generally any non-amenable graph, has positive speed, i.e. the distance from the origin grows linearly. Our focus is understanding whether permuted walks can be slower than the corresponding non-permuted walk, by carefully choosing the permutation sequence. We show that on regular trees (including the line), the permuted random walk is always stochastically faster. The proof relies on a majorization inequality for probability measures, plus an isoperimetric inequality for the tree. We also quantify how much slower the permuted random walk can possibly be when it is coupled with the corresponding non-permuted walk.Speeding up random walk mixing by starting from a uniform vertexhttps://zbmath.org/1536.054202024-07-17T13:47:05.169476Z"Díaz, Alberto Espuny"https://zbmath.org/authors/?q=ai:espuny-diaz.alberto"Morris, Patrick"https://zbmath.org/authors/?q=ai:morris.patrick-w"Perarnau, Guillem"https://zbmath.org/authors/?q=ai:perarnau.guillem"Serra, Oriol"https://zbmath.org/authors/?q=ai:serra.oriolSummary: The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks.
In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order \((\log n)^2\), speeding up the mixing to order \(\log n\). First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.Asymptotic capacity of the range of random walks on free products of graphshttps://zbmath.org/1536.054212024-07-17T13:47:05.169476Z"Gilch, Lorenz A."https://zbmath.org/authors/?q=ai:gilch.lorenz-aSummary: In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.Computation of Riesz \(\alpha\)-capacity \(\mathrm{C}_{\alpha}\) of general sets in \(\mathbb{R}^d\) using stable random walkshttps://zbmath.org/1536.310152024-07-17T13:47:05.169476Z"Nolan, John P."https://zbmath.org/authors/?q=ai:nolan.john-p"Audus, Debra J."https://zbmath.org/authors/?q=ai:audus.debra-j"Douglas, Jack F."https://zbmath.org/authors/?q=ai:douglas.jack-fSummary: A method for computing the Riesz \(\alpha\)-capacity, \(0 < \alpha \leq 2\), of a general set \(K \subset \mathbb{R}^d\) is given. The method is based on simulations of isotropic \(\alpha\)-stable motion paths in \(d\)-dimensions. The familiar walk-on-spheres method, often utilized for simulating Brownian motion, is modified to a novel walk-in-and-out-of-balls method adapted for modeling the stable path process on the exterior of regions ``probed'' by this type of generalized random walk. It accounts for the propensity of this class of random walk to jump through boundaries because of the path discontinuity. This method allows for the computationally efficient simulation of hitting locations of stable paths launched from the exterior of probed sets. Reliable methods of computing capacity from these locations are given, along with non-standard confidence intervals. Illustrative calculations are performed for representative types of sets \(K\), where both \(\alpha\) and \(d\) are varied.Reflected random walks and unstable Martin boundaryhttps://zbmath.org/1536.310332024-07-17T13:47:05.169476Z"Ignatiouk-Robert, Irina"https://zbmath.org/authors/?q=ai:ignatiouk-robert.irina"Kurkova, Irina"https://zbmath.org/authors/?q=ai:kurkova.irina-a"Raschel, Kilian"https://zbmath.org/authors/?q=ai:raschel.kilianSummary: We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp. non-rational), then the Martin boundary is countable, homeomorphic to \(\mathbb{Z} \cup \{\pm \infty \} \) (resp. uncountable, homeomorphic to \(\mathbb{R} \cup \{\pm \infty \} \)). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.Asymptotic behavior of solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delayshttps://zbmath.org/1536.352622024-07-17T13:47:05.169476Z"Cung The Anh"https://zbmath.org/authors/?q=ai:cung-the-anh."Vu Manh Toi"https://zbmath.org/authors/?q=ai:vu-manh-toi."Phan Thi Tuyet"https://zbmath.org/authors/?q=ai:phan-thi-tuyet.Summary: This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space \(BCL_{-\infty}(H)\). We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.The Clausius-Mossotti formulahttps://zbmath.org/1536.352652024-07-17T13:47:05.169476Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoineThis paper provides a proof of the Clausius-Mossotti formula for the effective conductivity in the dilute regime; here inclusions are allowed to overlap and that no upper bound is assumed on the number of points per unit volume. It is assumed that the shape of inclusions is spherical and the underlying point process assumes stationarity and ergodicity. The paper provides an estimate for the error that mostly comes from `interactions' between inclusions; the estimate relies on elliptic regularity theory. The approach is based on a short proof that the authors provided for Einstein's formula in a separate publication. This is a well-written paper with several details and it should be useful for someone working on dilute systems in many other contexts.
Reviewer: Vishnu Dutt Sharma (Ghandinagar)Quantum fractional Ornstein-Uhlenbeck semigroups and associated potentialshttps://zbmath.org/1536.353542024-07-17T13:47:05.169476Z"Ettaieb, Aymen"https://zbmath.org/authors/?q=ai:ettaieb.aymen"Missaoui, Sonia"https://zbmath.org/authors/?q=ai:missaoui.sonia"Rguigui, Hafedh"https://zbmath.org/authors/?q=ai:rguigui.hafedhSummary: Using an infinite-dimensional nuclear space, we introduce the quantum fractional number operator (QFNO) and the associated quantum fractional Ornstein-Uhlenbeck (O--U) semigroups. Then, we solve the Cauchy problems associated with the QFNO and show that its solutions can be expressed in terms of the aforementioned semigroups. Besides, we prove that the quantum fractional O--U semigroups satisfy the Feller property. Finally, using an adequate definition of the quantum fractional potentials, we give the solutions of the quantum fractional Poisson equations.The Dirichlet problem for Lévy-stable operators with \(L^2\)-datahttps://zbmath.org/1536.353912024-07-17T13:47:05.169476Z"Grube, Florian"https://zbmath.org/authors/?q=ai:grube.florian"Hensiek, Thorben"https://zbmath.org/authors/?q=ai:hensiek.thorben"Schefer, Waldemar"https://zbmath.org/authors/?q=ai:schefer.waldemarSummary: We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of \(2s\)-stable processes and exterior data, inhomogeneity in weighted \(L^2\)-spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit \(s\rightarrow1-\) which allows us to recover the local theory.Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projectionshttps://zbmath.org/1536.420052024-07-17T13:47:05.169476Z"Melentijević, Petar"https://zbmath.org/authors/?q=ai:melentijevic.petarSummary: In this paper we address the problem of finding the best constants in inequalities of the form:
\[
\big\| \big(|P_+ f|^s+|P_- f|^s \big)^{\frac{1}{s}} \big\|_{L^p(\mathbb{T})} \leq A_{p, s} \|f\|_{L^p (\mathbb{T})},
\]
where \(P_+ f\) and \(P_- f\) denote analytic and co-analytic projection of a complex-valued function \(f \in L^p (\mathbb{T}),\) for \(p \ge 2\) and all \(s>0\), thus proving Hollenbeck-Verbitsky conjecture from [\textit{B. Hollenbeck} and \textit{I. E. Verbitsky}, Oper. Theory: Adv. Appl. 202, 285--295 (2010; Zbl 1193.42064)]. We also prove the same inequalities for \(1 < p \leq \frac{4}{3}\) and \(s \leq \sec^2\frac{\pi}{2p}\) and confirm that \(s=\sec^2\frac{\pi}{2p}\) is the sharp cutoff for \(s\). The proof uses a method of plurisubharmonic minorants and an approach of proving the appropriate ``elementary'' inequalities that seems to be new in this topic. We show that this result implies best constants inequalities for the projections on the real-line and half-space multipliers on \(\mathbb{R}^n\) and an analog for analytic martingales. A remark on an isoperimetric inequality for harmonic functions in the unit disk is also given.\(G\)-stochastic maximum principle for risk-sensitive control problem and its applicationshttps://zbmath.org/1536.490272024-07-17T13:47:05.169476Z"Dassa, Meriyam"https://zbmath.org/authors/?q=ai:dassa.meriyam"Chala, Adel"https://zbmath.org/authors/?q=ai:chala.adelSummary: This study advances the \(G\)-stochastic maximum principle (\(G\)-SMP) from a risk-neutral framework to a risk-sensitive one. A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under \(G\)-Brownian motion (\(G\)-SDEs), where the control variable may influence all terms. We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost. Initially, we introduced an auxiliary process to address risk-sensitive performance costs within the \(G\)-expectation framework. Subsequently, we established and validated the correlation between the \(G\)-expected exponential utility and the \(G\)-quadratic backward stochastic differential equation. Furthermore, we simplified the \(G\)-adjoint process from a dual-component structure to a singular component. Moreover, we explained the necessary optimality conditions for this model by considering a convex set of admissible controls. To describe the main findings, we present two examples: the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.Noise sensitivity of random walks on groupshttps://zbmath.org/1536.600032024-07-17T13:47:05.169476Z"Benjamini, Itaï"https://zbmath.org/authors/?q=ai:benjamini.itai"Brieussel, Jérémie"https://zbmath.org/authors/?q=ai:brieussel.jeremieA random walk on a group \(G\) is a sequence of products \(X_n=s_1\ldots s_n\) of independent random variables \(s_i\) following identical distribution \(\mu\). The product \(X_n\) can be noised by resampling independently each increment \(s_i\) with probability \(\rho\in (0,1)\). This provides a new variable \(Y_n^\rho\) depending on \(X_n\). The random walk \((G,\mu)\) is \(l^1\)-noise sensitive if the law of the pair \((X_n,Y_n^\rho)\) and the law of the pair \((X_n,X_n^\prime)\) of two independent samples are close in the sense that their \(l^1\)-distance tends to zero. The random walk is entropy noise sensitive if the ratio between the conditional entropy \(H(Y_n^\rho|X_n)\) and \(H(X_n)\) tends to one. It is known that the space of bounded harmonic functions on \((G,\mu)\) is parametrized by the Poisson boundary. The random walk \((G,\mu)\) is Liouville if ths boundary is reduced to a point. In the present paper, the authors prove that groups with one of these two noise properties are necessarily Liouville. They also show that homomorphisms to free abelian groups provide an obstruction to \(l^1\)-sense sensitivity. Examples of \(l^1\) and entropy noise sensitive random walks are provided. Many open questions are also described at the end of the paper.
Reviewer: Ljuben Mutafchiev (Sofia)Exponential moments for disk counting statistics at the hard edge of random normal matriceshttps://zbmath.org/1536.600052024-07-17T13:47:05.169476Z"Ameur, Yacin"https://zbmath.org/authors/?q=ai:ameur.yacin"Charlier, Christophe"https://zbmath.org/authors/?q=ai:charlier.christophe"Cronvall, Joakim"https://zbmath.org/authors/?q=ai:cronvall.joakim"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatanSummary: We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let \(n\) be the number of points. We focus on two regimes: (a) the ``hard edge regime'' where all disk boundaries are at a distance of order \(\frac{1}{n}\) from the hard wall, and (b) the ``semi-hard edge regime'' where all disk boundaries are at a distance of order \(\frac{1}{\sqrt{n}}\) from the hard wall. As \(n\to +\infty\), we prove that the moment generating function enjoys asymptotics of the form
\[
\begin{aligned}
\exp \Big( C_1 n+C_2 \ln n+C_3 + \frac{C_4}{\sqrt{n}}+\mathcal{O}(n^{-\frac{3}{5}})\Big) \qquad \text{for the hard edge}, \\
\exp \Big( C_1 n+C_2 \sqrt{n}+C_3 +\frac{C_4}{\sqrt{n}}+\mathcal{O}\left(\frac{(\ln n)^4}{n}\right)\Big) \quad \text{for the semi-hard edge}.
\end{aligned}
\]
In both cases, we determine the constants \(C_1, \ldots, C_4\) explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the ``bulk'', ``soft edge'', and ``semi-hard edge'' regimes, the second and higher order cumulants of the disk counting function in the ``hard edge'' regime are proportional to \(n\) and not to \(\sqrt{n}\).Large gap asymptotics on annuli in the random normal matrix modelhttps://zbmath.org/1536.600062024-07-17T13:47:05.169476Z"Charlier, Christophe"https://zbmath.org/authors/?q=ai:charlier.christopheSummary: We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large \(n\) asymptotics of the form
\[
\exp \Bigg(C_1 n^2 + C_2 n \log n + C_3 n + C_4 \sqrt{n} + C_5 \log n + C_6 + {\mathcal{F}}_n + \mathcal{O} \Big(n^{-\frac{1}{12}}\Big)\Bigg),
\]
where \(n\) is the number of points of the process. We determine the constants \(C_1, \dots, C_6\) explicitly, as well as the oscillatory term \({\mathcal{F}}_n\) which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only \(C_1, \dots, C_4\) were previously known, (ii) when the hole region is an unbounded annulus, only \(C_1, C_2, C_3\) were previously known, and (iii) when the hole region is a regular annulus in the bulk, only \(C_1\) was previously known. For general values of our parameters, even \(C_1\) is new. A main discovery of this work is that \({\mathcal{F}}_n\) is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.The directed spanning forest in the hyperbolic spacehttps://zbmath.org/1536.600132024-07-17T13:47:05.169476Z"Flammant, Lucas"https://zbmath.org/authors/?q=ai:flammant.lucasSummary: The Euclidean directed spanning forest (DSF) is a random forest in \(\mathbb{R}^{d}\) introduced by \textit{F. Baccelli} and \textit{C. Bordenave} in [Ann. Appl. Probab. 17, No. 1, 305--359 (2007; Zbl 1136.60007)] and we introduce and study here the analogous tree in the hyperbolic space. The topological properties of the Euclidean DSF have been stated for \(d = 2\) and conjectured for \(d \geq 3\) (see further): it should be a tree for \(d \in \{2, 3\}\) and a countable union of disjoint trees for \(d \geq 4\). Moreover, it should not contain bi-infinite branches whatever the dimension \(d\). In this paper, we construct the hyperbolic directed spanning forest (HDSF) and we give a complete description of its topological properties, which are radically different from the Euclidean case. Indeed, for any dimension, the hyperbolic DSF is a tree containing infinitely many bi-infinite branches, whose asymptotic directions are investigated. The strategy of our proofs consists in exploiting the mass transport principle, which is adapted to take advantage of the invariance by isometries. Using appropriate mass transports is the key to carry over the hyperbolic setting ideas developed in percolation and for spanning forests. This strategy provides an upper-bound for horizontal fluctuations of trajectories, which is the key point of the proofs. To obtain the latter, we exploit the representation of the forest in the hyperbolic half space.Sectional Voronoi tessellations: characterization and high-dimensional limitshttps://zbmath.org/1536.600142024-07-17T13:47:05.169476Z"Gusakova, Anna"https://zbmath.org/authors/?q=ai:gusakova.anna"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophSummary: The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in \(\mathbb{R}^d\) with an \(\ell\)-dimensional affine subspaces, \(1 \leq \ell \leq d - 1\), are shown to be random tessellations of the same type but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation. The geometric properties of the typical cell and, more generally, typical \(k\)-faces, of the sectional Poisson-Voronoi tessellation are studied in detail. It is proved that in high dimensions, that is as \(d \to \infty \), the intersection of the \(d\)-dimensional Poison-Voronoi tessellation with an affine subspace of fixed dimension \(\ell\) converges to the \(\ell\)-dimensional Gaussian-Voronoi tessellation.Spectral telescope: convergence rate bounds for random-scan Gibbs samplers based on a hierarchical structurehttps://zbmath.org/1536.600182024-07-17T13:47:05.169476Z"Qin, Qian"https://zbmath.org/authors/?q=ai:qin.qian"Wang, Guanyang"https://zbmath.org/authors/?q=ai:wang.guanyangSummary: Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher-dimensional distributions to those targeting lower-dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain's spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an \(n\)-cube.Closure under infinitely divisible distribution roots and the Embrechts-Goldie conjecturehttps://zbmath.org/1536.600202024-07-17T13:47:05.169476Z"Xu, Hui"https://zbmath.org/authors/?q=ai:xu.hui.2"Yu, Changjun"https://zbmath.org/authors/?q=ai:yu.changjun"Wang, Yuebao"https://zbmath.org/authors/?q=ai:wang.yuebao"Cheng, Dongya"https://zbmath.org/authors/?q=ai:cheng.dongyaSummary: We show that the distribution class \(\mathcal{L}(\gamma)\setminus \mathcal{OS}\) is not closed under infinitely divisible distribution roots for \(\gamma >0\), that is, we provide some infinitely divisible distributions belonging to the class, whereas the corresponding Lévy distributions do not. In fact, one part of these Lévy distributions belonging to the class \(\mathcal{OL} \setminus \mathcal{L}(\gamma)\) have different properties, and the other parts even do not belong to the class \(\mathcal{OL}\). Therefore, combining with the existing related results, we give a completely negative conclusion for the subject and Embrechts-Goldie conjecture. Then we discuss some interesting issues related to the results of this paper.Multivariate stable approximation by Stein's methodhttps://zbmath.org/1536.600212024-07-17T13:47:05.169476Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng.3"Nourdin, Ivan"https://zbmath.org/authors/?q=ai:nourdin.ivan"Xu, Lihu"https://zbmath.org/authors/?q=ai:xu.lihu"Yang, Xiaochuan"https://zbmath.org/authors/?q=ai:yang.xiaochuanIn the setting of multivariate approximation by an \(\alpha\)-stable law, for some \(\alpha\in(0,2)\), using Stein's method, the authors establish new bounds on the solution to the underlying Stein equation which allow approximation results to be derived in Wasserstein-type distances. These rely on delicate density estimates for multivariate strictly stable laws. These bounds are then applied to yield rates of convergence in the classical multivariate stable limit theorem. Pareto-type examples are used to illustrate the sharpness of the rates obtained.
Reviewer: Fraser Daly (Edinburgh)Stein's method, smoothing and functional approximationhttps://zbmath.org/1536.600252024-07-17T13:47:05.169476Z"Barbour, A. D."https://zbmath.org/authors/?q=ai:barbour.andrew-d"Ross, Nathan"https://zbmath.org/authors/?q=ai:ross.nathan"Zheng, Guangqu"https://zbmath.org/authors/?q=ai:zheng.guangquSummary: Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals \(h\) of a càdlàg random process \(X\) of interest and the expectations of the same functionals of a well understood target random process \(Z\) with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we provide an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of \(X\) that is closely related to classical tightness criteria, a similar expectation for \(Z\), and, for the indicator of a set \(K\), the probability \(\mathbb{P}(Z \in K^{\theta} \setminus K^{-\theta})\) that the target process is close to the boundary of \(K\).Edgeworth expansions for volatility modelshttps://zbmath.org/1536.600272024-07-17T13:47:05.169476Z"Jirak, Moritz"https://zbmath.org/authors/?q=ai:jirak.moritzThe main contribution from this paper is to establish the validity of Edgeworth expansions both in the Kolmogorov and Wasserstein metric for various classes of popular volatility type models.
Reviewer: Romeo Negrea (Timişoara)On the quenched CLT for stationary Markov chainshttps://zbmath.org/1536.600282024-07-17T13:47:05.169476Z"Peligrad, Magda"https://zbmath.org/authors/?q=ai:peligrad.magdaLet \((\xi_i)_{i\in\mathbb{Z}}\) be a Markov chain with stationary measure \(\pi\). For \(f\) a measurable function with zero mean and finite second moment with respect to \(\pi\), the author considers the quenched central limit theorem for \(S_n=\sum_{i=1}^nf(\xi_i)\). That is, letting \(P^x\) denote the conditional probability given that the Markov chain is in state \(x\) at time zero, the author obtains necessary and sufficient conditions under which \(P^x(S_n/\sqrt{n}\leq t)\) converges to the distribution function of a Gaussian random variable for each \(t\) as \(n\to\infty\), for \(\pi\)-almost every \(x\). In particular, the author provides a new projective sufficient condition for the quenched central limit theorem. The proofs are based on conditioning the Markov chain with respect to both the past and the future.
Reviewer: Fraser Daly (Edinburgh)Moderate deviation principle for multiscale systems driven by fractional Brownian motionhttps://zbmath.org/1536.600292024-07-17T13:47:05.169476Z"Bourguin, Solesne"https://zbmath.org/authors/?q=ai:bourguin.solesne"Dang, Thanh"https://zbmath.org/authors/?q=ai:dang.thanh-ha|dang.thanh-tung|dang.thanh-hai|dang.thanh-hung|dang.thanh-n"Spiliopoulos, Konstantinos"https://zbmath.org/authors/?q=ai:spiliopoulos.konstantinos-vThe authors study the asymptotic behavior of the solution of the system
\[
dX_t^\varepsilon= g(X_t^\varepsilon,Y_t^\varepsilon)dt+\sqrt{\varepsilon}f(X_t^\varepsilon,Y_t^\varepsilon)dW_t^H, X_0^\varepsilon=x_0,
\]
\[
dY_t^\varepsilon=\frac{1}{\varepsilon}c(Y_t^\varepsilon)+ \frac{1}{\sqrt{\varepsilon}}\sigma(Y_t^\varepsilon)dB_t,Y_0^\varepsilon=y_0,
\]
where \(\varepsilon \rightarrow 0, t \in [0,1], (X_t^\varepsilon, Y_t^\varepsilon) \in \mathbb R^d\times \mathbb R^d,\) \(B\) is a standard Brownian motion and \(W^H\) is a \(p\)-dimensional fractional Brownian motion (fBm) with Hurst parameter \(H\in (1/2,1).\) Suppose that \(h(\varepsilon)\rightarrow \infty\) such that \(\sqrt{\varepsilon}h(\varepsilon)\rightarrow 0\) and let \(\bar X_t= \lim_{\varepsilon\rightarrow 0}X_t^\varepsilon.\) The moderate deviation process is defined by
\[
\eta_t^\varepsilon= \frac{X_t^\varepsilon- \bar X_t}{\sqrt{\varepsilon}h(\varepsilon)}.
\]
The authors investigate the asymptotic behaviour of the process \(X_t^\varepsilon\) as \(\varepsilon \rightarrow 0\) in the moderate deviation setting. It is shown that the resulting action functional is discontinuous in \(H\) at \(H=1/2\), indicating that the tail behaviour of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behaviour of such systems that are perturbed by standard Brownian motion.
Reviewer: B. L. S. Prakasa Rao (Hyderabad)Pathwise large deviations for the pure jump \(k\)-nary interacting particle systemshttps://zbmath.org/1536.600312024-07-17T13:47:05.169476Z"Sun, Wen"https://zbmath.org/authors/?q=ai:sun.wenThe author proves a pathwise large deviation result for the pure jump models of the \(k\)-nary interacting particle system introduced by Kolokoltsov that generalizes classical Boltzmann's collision model, Smoluchovski's coagulation model and others.
The upper bound is obtained by following the standard methods of using a process ``perturbed'' by a regular function. To show the lower bound, the author proposes a family of orthogonal martingale measures and proves a coupling for the general perturbations. The rate function is studied based on the idea of Léonard with a simplification by considering the conjugation of integral functionals on a subspace of \(L^{\infty}\). General ``gelling'' solutions in the domain of the rate function are also discussed.
Reviewer: Pavel Stoynov (Sofia)Optimal stopping of an Ornstein-Uhlenbeck bridgehttps://zbmath.org/1536.600392024-07-17T13:47:05.169476Z"Azze, A."https://zbmath.org/authors/?q=ai:azze.abel"D'Auria, B."https://zbmath.org/authors/?q=ai:dauria.bernardo"García-Portugués, E."https://zbmath.org/authors/?q=ai:garcia-portugues.eduardoSummary: In this paper we make a rigorous analysis of the existence and characterization of the free boundary related to the optimal stopping problem that maximizes the mean of an Ornstein-Uhlenbeck bridge. The result includes the Brownian bridge problem as a limit case. The methodology hereby presented relies on a time-space transformation that casts the original problem into a more tractable one with an infinite horizon and a Brownian motion underneath. We comment on two different numerical algorithms to compute the free-boundary equation and discuss illustrative cases that shed light on the boundary's shape. In particular, the free boundary generally does not share the monotonicity of the Brownian bridge case.Gambler's ruin with random stoppinghttps://zbmath.org/1536.600402024-07-17T13:47:05.169476Z"Morrow, Gregory J."https://zbmath.org/authors/?q=ai:morrow.gregory-jSummary: Let \(\{\mathbf{X}_j, j \geq 0\}\) denote a Markov process on \([-N-1, N+1] \cup \{\mathbf{c}\}\). Suppose \(\mathbb{P}(\mathbf{X}_{j+1} = m+1 | \mathbf{X}_j = m)=ph\), \(\mathbb{P}(\mathbf{X}_{j+1} = m-1 | \mathbf{X}_j = m)=(1-p)h\), all \(j \geq 1\) and \(|m| \leq N\), where \(p=\frac{1}{2}+\frac{b}{N}\) and \(h=1- c_N\) for \(c_N=\frac{1}{2} a^2/ N^2\). Define \(\mathbb{P}(\mathbf{X}_{j+1} = \mathbf{c} | \mathbf{X}_j = m) = c_N\), \(j \geq 0\), \(|m| \leq N\). \(\{\mathbf{X}_j\}\) terminates at the first \(j\) such that \(\mathbf{X}_j\in\{-N-1,N+1,\mathbf{c}\}\). Let \(\mathcal{L}=\max\{j\geq0: \mathbf{X}_j=0\}\). On \(\Omega\degree =\{\mathbf{X}_j\text{ terminates at }\mathbf{c}\}\), denote by \(\mathcal{R}\degree\) and \(\mathcal{L}\degree\), respectively, as the numbers of runs and steps from \(\mathcal{L}\) until termination. Denote \(\Delta\degree = \mathcal{L}\degree -2 \mathcal{R}\degree\). Then \(\lim_{N \to \infty}\mathbb{E}\{e^{\frac{i t}{N} \Delta\degree}| \Omega\degree \}= C_{a, b}\frac{\sqrt{c^2 + t^2} \big(\cosh \sqrt{c^2 + t^2} - \cosh (2 b)\big)}{(a^2 + t^2) \sinh \sqrt{c^2 + t^2}}\), where \(c^2= a^2+4 b^2\).Pedagogy of martingale convergence theoremhttps://zbmath.org/1536.600412024-07-17T13:47:05.169476Z"Nadkarni, M. G."https://zbmath.org/authors/?q=ai:nadkarni.mahendra-g"Bharani, Riddhi"https://zbmath.org/authors/?q=ai:bharani.riddhi(no abstract)Simple random walk on \(\mathbb{Z}^2\) perturbed on the axes (renewal case)https://zbmath.org/1536.600422024-07-17T13:47:05.169476Z"Andreoletti, Pierre"https://zbmath.org/authors/?q=ai:andreoletti.pierre"Debs, Pierre"https://zbmath.org/authors/?q=ai:debs.pierreThe authors study a simple random walk on \(\mathbb{Z}^2\) with constraints on the axes. The model under consideration is partly inspired by an experiment in physics when a quasi-two dimensional cloud of cold neutral atoms is submitted to a pressure forces exerted by a laser beam. The authors simplify the model in many directions, namely, they consider a single particle instead of a system of particles (the gas of atoms), and take a simple form for the applied force which decreases when the particle is getting closer to the origin. This simulates that at a certain moment the pressure exerted by the force is not enough to confine the particle in a single point. It is assumed that a particle evolves freely in the cones but when touching the axes a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axes. The main ideas of the proof of this result are carefully explained, which makes the paper convenient for the reader.
Reviewer: Yuliya S. Mishura (Kyïv)A symmetric random walk on the vertices of a hexahedronhttps://zbmath.org/1536.600432024-07-17T13:47:05.169476Z"Sarkar, Jyotirmoy"https://zbmath.org/authors/?q=ai:sarkar.jyotirmoySummary: A symmetric random walk on the vertices of a hexahedron (cube) starts at a vertex called the origin; and at each step it moves, to an adjacent vertex with probability 1/3 each. We find the means and the standard deviations of the number of steps needed to (1) return to origin, (2) visit all vertices, and (3) return to origin after visiting all vertices. We also find (i) the number of vertices visited before return to origin, (ii) the last vertex visited, and (iii) the number of vertices visited while returning to origin after visiting all vertices.An integration by parts formula for functionals of the Dirichlet-Ferguson measure, and applicationshttps://zbmath.org/1536.600442024-07-17T13:47:05.169476Z"Flint, Ian"https://zbmath.org/authors/?q=ai:flint.ian"Torrisi, Giovanni Luca"https://zbmath.org/authors/?q=ai:torrisi.giovanni-lucaSummary: The Dirichlet-Ferguson measure is a random probability measure that has seen widespread use in Bayesian nonparametrics. Our main results can be seen as a first step towards the development of a stochastic analysis of the Dirichlet-Ferguson measure. We define a gradient that acts on functionals of the measure and derive its adjoint. The corresponding integration by parts formula is used to prove a covariance representation formula for square integrable functionals of the Dirichlet-Ferguson measure and to provide a quantitative central limit theorem for the first chaos. Our findings are illustrated by a variety of examples.Reflected BSDEs with logarithmic growth and applications in mixed stochastic control problemshttps://zbmath.org/1536.600472024-07-17T13:47:05.169476Z"El Asri, Brahim"https://zbmath.org/authors/?q=ai:el-asri.brahim"Oufdil, Khalid"https://zbmath.org/authors/?q=ai:oufdil.khalidIn this article, the authors study the existence and the uniqueness of a solution for reflected backward SDEs in the case when the generator is logarithmic growth in the $z$-variable $(|z|\sqrt{| \ln(|z|)|)}$, the terminal value and obstacle are an $L^p$-integrable, for a suitable $p>2$. To construct the solution they use localization method, and also apply these results to get the existence of an optimal control strategy for the mixed stochastic control problem in finite horizon.
Reviewer: Kai Wang (Bengbu)Approximation of the solution of stochastic differential equations driven by multifractional Brownian motionhttps://zbmath.org/1536.600492024-07-17T13:47:05.169476Z"Soós, Anna"https://zbmath.org/authors/?q=ai:soos.anna(no abstract)Stochastic wave equation with Lévy white noisehttps://zbmath.org/1536.600502024-07-17T13:47:05.169476Z"Balan, Raluca"https://zbmath.org/authors/?q=ai:balan.raluca-mAn equation \[ \partial_{tt}u=\Delta u+\sigma(u)\,\dot L \] is considered on \(\mathbb R^d\) for \(d=1,2\), with deterministic initial conditions \(u(0)=u_0\) and \(\partial_tu(0)=v_0\). Here \(\sigma\) is a Lipschitz continuous real valued function on \(\mathbb R\), \(L\) has the form \[ L(A)=b|A|+\int_{A\times\{|z|\le 1\}}z\widetilde J(dt,dx,dz)+\int_{A\times\{|z|>1\}}zJ(dt,dx,dz), \] where \(A\subseteq\mathbb R_+\times\mathbb R^d\) is of finite Lebesgue measure \(|A|\), \(b\in\mathbb R\), \(J\) is a Poisson random measure on \(\mathbb R_+\times\mathbb R^d\times\mathbb R\) of intensity \(dtdx\nu(dz)\), \(\widetilde J\) is the compensated version of \(J\) and \(\nu\) is a Lévy measure on \(\mathbb R\). In addition, \(u_0\) is assumed to be bounded and continuous and \(v_0\) bounded and measurable if \(d=1\), and \(u_0\in C^1(\mathbb R^2)\cap W^{1,q}(\mathbb R^2)\) and \(v_0\in L^q(\mathbb R^2)\) for some \(q>2\) if \(d=2\). It is proved that there exists a unique solution provided that there exists \(0<q\le 2\) such that \[ \int_{\{|z|>1\}}|z|^q\,d\nu<\infty \] if \(d=1\), or there exist \(0<q\le p<2\) and \[ \int\min\{|z|^p,|z|^q\}\,d\nu<\infty \] if \(d=2\). Moreover, if \(d=2\) then the solution has càdlàg paths in \(H^r_{\mathrm{loc}}(\mathbb R^2)\) for \(r<-1\), and if \(d=1\), \(u_0,v_0\in L^1(\mathbb R)\) and \(\widehat u_0\) grows at most as \((1+|\xi|^2)^{-1}\) then the solution has càdlàg paths in \(H^r_{\mathrm{loc}}(\mathbb R)\) for \(r<1/4\).
Reviewer: Martin Ondreját (Praha)Comparison principle for stochastic heat equations driven by \(\alpha \)-stable white noiseshttps://zbmath.org/1536.600562024-07-17T13:47:05.169476Z"Wang, Yongjin"https://zbmath.org/authors/?q=ai:wang.yongjin"Yan, Chengxin"https://zbmath.org/authors/?q=ai:yan.chengxin"Zhou, Xiaowen"https://zbmath.org/authors/?q=ai:zhou.xiaowenSummary: For a class of non-linear stochastic heat equations driven by \(\alpha \)-stable white noises for \(\alpha \in(1, 2)\) with Lipschitz coefficients, we prove the existence and pathwise uniqueness of \(L^p\)-valued càdlàg solution to such an equation for \(p \in(\alpha, 2]\) by considering a sequence of approximating stochastic heat equations driven by truncated \(\alpha \)-stable white noises obtained by removing the big jumps from the original \(\alpha \)-stable white noise. If the \(\alpha \)-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we further prove a comparison theorem on the \(L^2\)-valued càdlàg solutions to such an equation. As a consequence, the non-negativity of the \(L^2\)-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.On bivariate distributions of the local time of Itô-McKean diffusionshttps://zbmath.org/1536.600692024-07-17T13:47:05.169476Z"Jakubowski, Jacek"https://zbmath.org/authors/?q=ai:jakubowski.jacek"Wiśniewolski, Maciej"https://zbmath.org/authors/?q=ai:wisniewolski.maciejThe problem of finding the joint distribution of a couple of processes \((X_t,L_t)\), where \(X\) is an Itô-McKean diffusion process and \(L\) its local time at 0, motivated the research of the authors reported in this article.
The authors produce a new explicit representation of the bivariate distribution of \((X_t,L_t)\), where \(X\) is an Itô-McKean diffusion process and \(L\) its local time at 0. The key idea is the use of excursion theory in two respects: first, it allows one to obtain the transition density of the couple \((X,L)\), secondly, they get a simple connection formula for the distribution of excursions from a hyperplane of the couple \((X,L)\). Formulas are obtained for several functionals of \(L\), expressed in terms of the transition density \(p\) with respect to the velocity (speed) measure, recalling that the Green kernel for the above class of diffusions is
\[G_\lambda(x, z)=\int_0^{\infty} e^{-\lambda t} p_t(x, z) d t, \quad \lambda>0, x, z \in E,\]
where \(E\subset\mathbb{R}\) contains \(0\) and this is a boundary and instantaneous reflecting point.
The paper is organized into three sections and various subsections. After a well-documented historical review on the subject, covered in the introductory section, the presentation of the theoretical core is given in Section two, finishing the paper in the third one, with the rigorous proofs of all new results.
This is an interesting paper, the main features of which are summarized here.
First, there is a new explicit description of the distribution of \(L_t\) in terms of a convolution exponent. This is done in subsection 2.1. The probability distribution \(Q(t,z)\) of \(L_t\) for an Itô-McKean diffusion starting from 0, with transition density \(p\) as before satisfies the equation
\[p * \frac{\partial Q}{\partial z}+Q-1=0.\]
The authors characterize in Theorem 2.1 the class of functions that solves the above equation, which provides an important tool to determine \(Q\). This is the first step to provide a convolution representation of the density of \(L_t\) for fixed \(t>0\). This is an integral representation in terms of a Bessel function and the transition probability. To do that, they define a convolution exponent as follows. Let \(\mathcal{A}\) be the set of all complex-valued locally integrable functions on \([0, \infty)\). For any \(f \in \mathcal{A}\), the authors define
\[
\mathcal{E}^*(f)=\delta_0+\sum_{i \geq 1} \frac{f^{* i}}{i !}.
\]
Then an ingenious use of Bessel functions together with the introduction of a function \(\varphi (t,s)=1*\mathcal{E}^*(-tp)(s)\) allows the authors to prove Theorem 2.4 giving the distribution of \(L_t\).
The paper further explores the excursion theory of Markov processes to determine the distribution of \((X_t, L_tt)\) and the excursions from a hyperplane for a bivariate diffusion. So, a description of the transition density of the pair \((X, L)\) using excursion theory is given in subsection 2.2, through Theorems 2.7 and 2.8, where convolution and class \(\mathcal{A}\) again play a major role.
Further, the paper shows a simple connection formula for the excursion distribution of a bivariate Itô-McKean diffusion from a hyperplane in subsection 2.3 (see Theorem 2.9 and Corollary 2.10).
Examples include a local time distribution and a formula for the distribution of \((X_t, L_\infty)\) for transient diffusion (Theorem 2.15). Other examples of applications of local-time bivariate distributions are presented, including a probabilistic representation of the solution to the generalized Stroock-Williams equation. The article also mentions previous studies on the probabilistic structure of local time \(L\) for a 0-reflected Itô diffusion.
Processes that meet the assumptions of the study include squared Bessel processes with index \(\mu\in (-1,0)\), radial Ornstein-Uhlenbeck processes, or Pearson diffusion.
Reviewer: Rolando Rebolledo Berroeta (Santiago de Chile)Freezing limits for Calogero-Moser-Sutherland particle modelshttps://zbmath.org/1536.600732024-07-17T13:47:05.169476Z"Voit, Michael"https://zbmath.org/authors/?q=ai:voit.michaelSummary: One-dimensional interacting particle models of Calogero-Moser-Sutherland type with \(N\) particles can be regarded as diffusion processes on suitable subsets of \(\mathbb{R}^N\) like Weyl chambers and alcoves with second-order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman-Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman-Opdam Jacobi polynomials, and Heckman-Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like \(\beta \)-Hermite, \( \beta \)- Laguerre, and \(\beta \)-Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so-called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed \(N\), one or several of the coupling parameters tend to \(\infty \). In many cases, the limits will be \(N\)-dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one-dimensional orthogonal polynomials of order \(N\).
{\copyright} 2023 The Authors. \textit{Studies in Applied Mathematics} published by Wiley Periodicals LLC.Uniqueness of first passage time distributions via Fredholm integral equationshttps://zbmath.org/1536.600752024-07-17T13:47:05.169476Z"Christensen, Sören"https://zbmath.org/authors/?q=ai:christensen.soren-gram|christensen.soren-torholm|christensen.soren.2|christensen.soren.1"Fischer, Simon"https://zbmath.org/authors/?q=ai:fischer.simon"Hallmann, Oskar"https://zbmath.org/authors/?q=ai:hallmann.oskarThe authors provide a proof for the uniqueness of first-passage time of a Brownian motion. The proof of the uniqueness relies on the dominated convergence theorem of integration. This is a crucial point in the paper, which is actually innovative. First passage times are applicable in any field of the role for Brownian motion, like actuarial science and finance. This is a reason for an additional importance of the paper.
Reviewer: Christos E. Kountzakis (Karlovassi)The trunks of \(\mathrm{CLE}(4)\) explorationshttps://zbmath.org/1536.600772024-07-17T13:47:05.169476Z"Lehmkuehler, Matthis"https://zbmath.org/authors/?q=ai:lehmkuehler.matthisThe core of the present paper concerns a continuity result for the laws of \(\mathrm{CLE}(4)\) exploration paths and their trunks when \(k\) varies \(\mathrm{CLE}(4)\) (conformal loop ensemble, i.e., a random collection of disjoint simple loops in a simply connected domain in the plane which is of particular interest in random geometry). First, the author analyzes the case of the trunks and then he studies the exploration paths themselves. Finally, the pieces are put together and concludes the proof of the main result (Theorem 1.2).
Reviewer: Romeo Negrea (Timişoara)Exponential ergodicity for stochastic equations of nonnegative processes with jumpshttps://zbmath.org/1536.600842024-07-17T13:47:05.169476Z"Friesen, Martin"https://zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Kremer, Jonas"https://zbmath.org/authors/?q=ai:kremer.jonas"Rüdiger, Barbara"https://zbmath.org/authors/?q=ai:rudiger.barbaraThis article studies the long-time behavior of a large class of continuous-time Markov processes on the state space \(\mathbb{R}_{\geq 0} := [0, \infty)\), constructed as strong solutions to stochastic differential equations with jumps. Under a global dissipativity condition combined with mild conditions on the parameters of the SDE, they show these processes to be exponentially ergodic for various Wasserstein distances on \(\mathbb{R}_{\geq 0}\). This result is then showed to apply to continuous-state branching processes with immigration (CBI), to non-linear CBI processes, to the \(Q\)-process of a CBI and to CBI processes in Lévy random environments.
Reviewer: Bastien Mallein (Toulouse)Lipschitzian norms and functional inequalities for birth-death processeshttps://zbmath.org/1536.600862024-07-17T13:47:05.169476Z"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.10This paper treats a birth-death process with generator \({\mathcal L}\) and reversible invariant probability measure \(\pi\). The author identifies explicitly the Lipschitzian norm of the solution of the Poisson equation \(- {\mathcal L} G = g - \pi(g)\) for \(\vert g \vert \leqslant \varphi\). This leads to some transportation-information inequalities, concentration inequalities and Cheeger-type isoperimetric inequalities. Lastly, several examples are provided to illustrated the results.
More precisely, let \(( X_t )_{ t \geq 0 }\) be a birth-death process on \({\mathbb N} = \{ 0, 1, 2, \dots \}\) with birth rates \( ( b_i )_{ i \in {\mathbb N} }\) and death rates \(( a_i )_{ i \in {\mathbb N} }\), i.e., its generator \({\mathcal L}\) is given for any real function \(G\) on \({\mathbb N}\) by
\[ {\mathcal L} G(i) = b_i ( G( i+1) - G(i) ) + a_i ( G( i-1) - G(i)),
\tag{1}\]
where \(b_i\) and \(a_i\) are positive for any \(i \geq 1\), with furthermore \(b_0 > 0\) and \(a_0 = 0\). For any real function \(G\), \(G( -1)\) is identified as \(G(0)\) by convention. In this paper, the author always assumes that the process is positive recurrent, i.e.,
\[ \sum_{ n \geq 0} \mu_n \sum_{ i \geq n} ( \mu_i b_i )^{-1} = \infty \qquad \text{and} \qquad C := \sum_{n=0}^{+ \infty} \mu_n < + \infty,
\tag{2}\]
where \(\mu\) given by
\[ \mu_0 = 1, \qquad \mu_n = \frac{ b_0 b_1 \cdots b_{n-1} }{ a_1 a_2 \cdots a_n}, \qquad n \geq 1
\tag{3}\]
is an invariant measure of the process. Define the normalized probability measure \(\pi\) of \(\mu\) by \(\pi_n = \mu_n / C\) for any \(n \geq 0\), which is actually the reversible invariant probability measure of the process. Given an increasing function \(\rho : {\mathbb N} \to {\mathbb R}\), define \(d_{\rho} (i,j)\) \(=\) \(\vert \rho(i) - \rho(j) \vert\) a metric on \({\mathbb N }\) with respect to \(\rho\). We say that a function \(G\) on \({\mathbb N}\) is Lipschitz with respect to \(\rho\) (or \(\rho\)-Lipschitz), if
\[
\begin{aligned}
\Vert G \Vert_{ Lip ( \rho) } &:= \sup_{ i \not= j} \frac{ \vert G(j) - G(i) \vert}{ \vert \rho(j) - \rho(i) \vert} \\
& = \sup_{ i \geq 0} \frac{ \vert G(i+1) - G(i) \vert}{ \rho( i+1) - \rho(i) } < + \infty.
\end{aligned} \tag{4}
\]
The space of all \(\rho\)-Lipschitz functions is denoted by \(C_{ Lip( \rho) }\). Throughout this paper, we assume that \(\rho \in L^1(\pi)\) and denote by \(( C_{ Lip( \rho)}^0, \Vert \cdot \Vert_{ Lip( \rho)} )\) the space of all \(\rho\)-Lipschitz functions with \(\pi(G)\) \(:=\) \(\int G d \pi\) \(= 0\). Consider the Poisson equation \(- {\mathcal L} G = g\). Recall the usual Lipschitzian norm of \(( - {\mathcal L} )^{-1}\) on \(C_{ Lip( \rho)}^0\):
\[ \Vert ( - {\mathcal L} )^{-1} \Vert_{ Lip( \rho )} := \sup \{ \Vert ( - {\mathcal L} )^{-1} g \Vert_{ Lip( \rho)} : \quad g \in C_{ Lip(\rho)}^0, \quad \Vert g \Vert_{ Lip( \rho)} \leqslant 1 \}.
\tag{5}\]
By definition, we say that \({\mathcal L}\) has a spectral gap in \(C_{ Lip( \rho)}^0\) if \(0\) is an isolated eigenvalue of \(- {\mathcal L}\) in \(C_{ Lip( \rho)}^0\), or equivalently \(( \- {\mathcal L} )^{-1}\) : \( C_{ Lip( \rho)}^0\) \(\mapsto\) \(C_{ Lip( \rho)}^0\) is bounded. Let \(\lambda_1\) be the spectral gap of \(- {\mathcal L}\) in \(L^2( \pi)\), i.e., the infimum of the spectrum of \(- { \mathcal L}\) in \(L^2(\pi)\). In this paper, the author forcuses on the observed function \(g\) which is bounded by some nonnegative function \(\varphi\) but not \(\rho\)-Lipschitz continuous. The aim of this paper is to get the concentration inequalities for the empirical mean \(1 / t \int_0^t g(X_s) ds\) through non-Lipschitz observables. Indeed, these concentration inequalities are immediately the consequences of the estimation on \(\Vert G \Vert_{ Lip( \rho)}\) via the martingale decomposition or transportation-information inequalities. For this purpose the author first calculates directly
\[ \sup_{ \vert g \vert \leqslant \varphi} \Vert ( - {\mathcal L} )^{-1} ( g - \pi(g) ) \Vert_{ Lip( \rho)}
\tag{6}\]
since the solution of Poisson equation \(- {\mathcal L} G = g\) for birth-death process can be solved explicitly. Then the author can get the transportation-information inequalities by the boundedness of \(\Gamma( \rho)\) or the Lyapunov test function approach. Especially, by taking
\[ d_{\varphi} (i,j) := ( \varphi(i) + \varphi(j) ) \mathbf{1}_{ i \not= j},
\tag{7}\]
the Wasserstein distance \(W_{1, d_{\varphi} } (\nu, \mu)\) becomes the weighted total variation distance \(\Vert \varphi ( \nu - \mu ) \Vert_{TV}\). The author also obtains some Cheeger-type isoperimetric inequalities, which can lead to transportation-information inequalities and concentration inequalities under some additional assumptions.
Let us enumerate below the main results of the present article. We define the carré-du-champ operator \(\Gamma\) of birth-death process with respect to \({\mathcal L}\) as follows:
\[ \Gamma(f,g) := \frac{1}{2} [ {\mathcal L} (f g) - {\mathcal L}f g - f {\mathcal L}g ],
\tag{8}\]
where \(f\) and \(g\) are any functions on \({\mathbb N}\). Note that \(\Gamma (f,f) = \Gamma(f)\). The Dirichlet form associated with \({\mathcal L}\) is given by any functions \(f\) and \(g\) on \({\mathbb N}\)
\[ {\mathcal E} (f,g) := \langle - {\mathcal L} f, g \rangle_{ L^2( \pi)} = \sum_{k=0}^{ + \infty} \pi_k \Gamma(f,g) (k),
\tag{9}\]
with \({\mathcal E} (f,f) = {\mathcal E}(f)\). Let \({\mathcal M}_1\) be the space of probability measures on \({\mathbb N}\). For all \(\mu, \nu \in {\mathcal M}_1\), the Fisher-Donsker-Varadhan's information of \(\nu\) with respect to \(\mu\) is defined by
\[ I( \nu \vert \mu ) := \begin{cases} {\mathcal E} ( \sqrt{f}, \sqrt{f} ), &\text{if} \quad \nu = f \mu, \\
+ \infty, &\quad \text{otherwise}. \end{cases}
\tag{10}\]
The Wasserstein distance between \(\nu\) and \(\mu\) with respect to a given metric \(d\) on \({\mathbb N}\) is defined by
\[ W_{1, d}(\nu, \mu ) = \inf_{\gamma} \sum_{i=0}^{+ \infty} \sum_{j=0}^{ + \infty} \gamma_{ij} \cdot d(i, j),
\tag{11}\]
where \(\gamma\) runs over all couplings of \(\nu\) and \(\mu\), i.e., all probability measures \(\gamma\) on \({\mathbb N}^2\) with marginal distribution \(\nu\) and \(\mu\).
Theorem 1. Let \({\mathcal A}\) be the set of all real increasing functions \(\rho\) on \({\mathbb N}\) such that \(\rho \in L^1( \pi)\). Let \(\rho \in {\mathcal A}\) and \(\varphi\) be a nonnegative function in \(L^1( \pi)\) such that \(c(\varphi, \rho) < + \infty\). If there exists a positive constant \(M\) such that \(\Gamma(\rho) (k) \leqslant M\), \(\text{ for all } k \in {\mathbb N}\), then for all \(\nu \in {\mathcal M}_1\), we have the following transportation-information inequality
\[ W_{1, d_{\varphi} } (\nu, \pi) \leqslant 2 c(\varphi, \rho) \sqrt{ M \cdot I(\nu \vert \pi) },
\tag{12}\]
where \(d_{\varphi}\) is the metric on \({\mathbb N}\) defined by (8).
Instead of using the boundedness of \(\Gamma\), we also have the following generalized transportation-information inequality by Lyapunov test function condition.
Theorem 2. Let \(\rho\) and \(\varphi\) be the same as in Theorem 1. Assume that the following Lyapunov condition holds: for some \(\delta > 0\) there exists a function \(V\) : \({\mathbb N} \to [1, \infty)\), \(V \in L^1( \pi)\) such that for any \(k \in {\mathbb N}\)
\[ ( 1 + \delta ) a_k ( \rho (k-1) - \rho (k) )^2 + \left( 1 + \frac{1}{\delta} \right) b_k ( \rho(k+1) - \rho(k) )^2 \leqslant - \alpha \frac{ {\mathcal L} V }{V} (k) + \beta,
\tag{13}\]
where \(\alpha\) and \(\beta\) are two positive constants. Then for all \(\nu \in {\mathcal M}_1\), we have
\[ W_{1, d_{\varphi} } ( \nu, \pi) = \Vert \varphi ( \nu - \pi) \Vert_{TV} \leqslant c( \varphi, \rho) \sqrt{ \alpha I^2(\nu \vert \pi) + \beta I( \nu \vert \pi) }.
\tag{14}\]
Let \(c_{ch}\) be the best constant in the following kind of Cheeger-type isoperimetric inequality for birth-death process, i.e., for any \(f \in L^1( \pi)\)
\[ \sum_{ k=0}^{+ \infty} \pi_k \vert f(k) - \pi(f) \vert \leqslant c_{ch} \sum_{ k=0}^{+ \infty} \pi_k b_k \vert f( k+1) - f(k) \vert.
\tag{15}\]
Then we have \(c_{ch} < c( 1, \rho_0)\).
Theorem 3. If \(c_{ch} < + \infty\) and there exists a positive constant \(M\) such that if \(a_k + b_k \leqslant M\), \((\text{ for all } k \in {\mathbb N})\), then for all \(\nu \in {\mathcal M}_1\) we have the following transportation-information inequality
\[ W_{i, d_1} ( \nu, \pi) = \Vert \nu - \pi \Vert_{TV} \leqslant c_{ch} \sqrt{ 2 M} \sqrt{ I( \nu \vert \pi ) }.
\tag{16}\]
Or equivalently, for every function \(g\) on \({\mathbb N}\) such that \(\vert g \vert \leqslant 1\), we have for any initial measure \(\nu \ll \pi\) and \(t, r > 0\),
\[ {\mathbb P}_{\nu} \left( \frac{1}{t} \int_-^t g(X_s) ds - \pi(g) > r \right) \leqslant \left\Vert \frac{ d \nu}{ d \pi} \right\Vert_{ L^2(\pi)} \exp \left\{ - \frac{ t r^2}{ 2 M c_{ch}^2 } \right\}.
\tag{17}\]
For other related works, see, e.g., [\textit{A. Guillin} et al., Probab. Theory Relat. Fields 144, No. 3--4, 669--695 (2009; Zbl 1169.60304)] for transportation-information inequalities for Markov processes, [\textit{W. Liu} and \textit{Y. Ma}, Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 1, 58--69 (2009; Zbl 1172.60023)] for spectral gap and convex concentration inequalities for birth-death processes, and [\textit{Y. Ma} et al., Electron. Commun. Probab. 16, 600--613 (2011; Zbl 1254.60027)] for transportation-information inequalities for continuous Gibbs measures.
Reviewer: Isamu Dôku (Saitama)Maximal displacement of spectrally negative branching Lévy processeshttps://zbmath.org/1536.600872024-07-17T13:47:05.169476Z"Profeta, Christophe"https://zbmath.org/authors/?q=ai:profeta.christopheSummary: We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particle. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards \(+ \infty \), and is exponential when it drifts towards \(- \infty \).Equality of critical parameters for percolation of Gaussian free field level setshttps://zbmath.org/1536.600932024-07-17T13:47:05.169476Z"Duminil-Copin, Hugo"https://zbmath.org/authors/?q=ai:duminil-copin.hugo"Goswami, Subhajit"https://zbmath.org/authors/?q=ai:goswami.subhajit"Rodriguez, Pierre-François"https://zbmath.org/authors/?q=ai:rodriguez.pierre-francois"Severo, Franco"https://zbmath.org/authors/?q=ai:severo.francoIn this paper, the authors consider upper level sets of Gaussian free fields in $\mathbb Z^d$ for $d>2$ above a given height \(h\) and relates this model with a strongly correlated (algebraic decay) canonical percolation model. They introduce a new renormalization framework used in an interpolation scheme that allows to rigorously integrate out the long-range dependence of the GFF and identify relevant critical parameters of the model(s), in general percolation models that do not necessarily enjoy the usual finite-energy properties.
Authors' abstract: We consider upper level sets of the Gaussian free field (GFF) on \({\mathbb{Z}^d}\), for \(d\ge 3\), above a given real-valued height parameter \(h. As h\) varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated with this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the GFF. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.
Reviewer: Arnaud Le Ny (Paris)Extremal regime for one-dimensional Mott variable-range hoppinghttps://zbmath.org/1536.600952024-07-17T13:47:05.169476Z"Croydon, David A."https://zbmath.org/authors/?q=ai:croydon.david-a"Fukushima, Ryoki"https://zbmath.org/authors/?q=ai:fukushima.ryoki"Junk, Stefan"https://zbmath.org/authors/?q=ai:junk.stefanThe paper studies the asymptotic behaviour of a version of the Mott random walk in dimension one. This process describes the movement of a particle in a random environment and arises in physics, specifically as the dynamics of an electron in a disordered conduction medium. The process is designed to model variable range hopping (VRH), under which the electron moves between localised states over varying distances.
The process is defined in terms of the realisation \(\omega\) of a unit-intensity homogeneous Poisson process on the real line (the assumption of unit-intensity is for notational convenience and could be removed), conditioned to have an atom at 0 and labelled such that \(\cdots<\omega_{-2}<\omega_{-1}<\omega_0=0<\omega_1<\omega_2<\cdots\). Given \(\omega\), the Mott random walk is a continuous-time Markov chain that jumps from site \(i\) to site \(j\) at rate proportional to
\begin{align*}
c^{\alpha,\lambda}(\omega_i,\omega_j)=\exp\{-|\omega_i-\omega_j|^\alpha + \lambda(\omega_i+\omega_j) \}.
\end{align*}
Thus, there is a drift term defined in terms of the constant \(\lambda\in\mathbb{R}\) and the variable-range hopping term with exponent \(\alpha>1\).
In the paper, three main results are proved. The first states that asymptotically, taking the drift to zero, the Mott random walk is situated between two environment-measurable barriers. The second gives the scaling limit (under the annealed law) of the running supremum and infimum processes. The third states that the asymptotic finite-dimensional distributions of the Mott random walk has density proportional to \(e^{2\lambda x}\) between the barriers obtained in the first main result.
To state the results more precisely, define the running maxima and minima process \[ \overline X_t = \sup_{s\leq t} X_s,\quad\quad \underline{X}_t = \inf_{s\leq t} X_s, \] and their right-continuous inverses \(\Delta_x^+\) and \(\Delta_x^-\). The first main result of the paper states that if \(d_{M_1}\) metrises the local \(M_1\) Skorokhod topology then as \(n\to\infty\), \[ d_{M_1}\left( \left( \frac1n \underline {X}_{nL^{-1}(nt)} , \frac1n \overline {X}_{nL^{-1}(nt)} \right)_{t\geq 0}, \left(m_{n,-}^{-1}(t) , m_{n,+}^{-1}(t)\right)_{t\geq 0} \right) \longrightarrow 0, \] as \(n\to\infty\), in probability under the annealed law of the Mott random walk with VRH exponent \(\alpha\) and drift \(\lambda/n\) (i.e., the drift tends to zero with \(n\)). Here, \(m^{-1}_{n+}(x)\) and \(m^{-1}_{n-}\) are explicitly defined functions of the environment \(\omega\) and the function \(L(u)=\exp\{\log^{1/\alpha}(u)\}\) is slowly varying. An analogous convergence result holds for the right-continuous inverses \(\Delta_x^+\) and \(\Delta_x^-\).
The second main result characterises explicitly the \(n\to\infty\) limit, in probability, of \(m^{-1}_{n+}(x)\) and \(m^{-1}_{n-}\) and their right-continuous limits: Let \(\{x_i,v_i\}\) be the atoms of a Poisson process on \(\mathbb{R}\times (0,\infty)\) of intensity \(\nu^{-2}\,d\nu\,dx\), then \(m^{-1}_{n+}\) and \(m^{-1}_{n-}\) converge jointly in probability to the right-continuous inverses of \(m_+\) and \(m_-\) given by \[ m_+(x)=\sup\{\nu_i\colon 0\leq x_i\leq x\},\quad\quad m_-(x)=\sup\{\nu_i\colon -x\leq x_i\leq 0\}. \]
The third result of the paper concerns the asymptotic finite-dimensional distributions of the Mott random walk: firstly, for continuous bounded functions \(f_1,\ldots,f_k\) and \(0<t_1<\ldots<t_k\), we have the following convergence in probability (with respect to the law of the environment), \[ \left| E_\omega^{\alpha,\lambda/n} \left[\prod_{i=1}^k f_i\left( \frac1n X_{nL^{-1}(nt_i}) \right) \right] - \prod_{i=1}^k \frac{\int e^{2\lambda x}f_i(x)\, dx}{\int e^{2\lambda x}\, dx} \right| \longrightarrow 0 \quad\text{as }n\to\infty, \] where \(E_\omega\) denotes expectation with respect to the quenched law of the walk (i.e., conditioned on a given realisation of the Poisson process \(\omega\)) and the integrals in the fraction are from \(m_{n,-}^{-1}(t_i)\) to \(m_{n,+}^{-1}(t_i)\). Finally, under the annealed law \(\mathbb{P}^{\alpha, \lambda/n}\) (i.e., the law of the walk integrated out over the environment), the random vector \((n^{-1}X_{nL^{-1}(nt_1)},\ldots,n^{-1}X_{nL^{-1}(nt_k)})\) converges to \((U_{t_1}^\lambda,\ldots U_{t_k}^\lambda)\), an independent collection of random variables such that, conditional on \((m_-,m_+)\), the law of \(U_{t_i}^\lambda\) has density \(e^{2\lambda x}\) on \([m_-^{-1}(t_i), m_+^{-1}(t_i)]\) and zero elsewhere.
Reviewer: Janosch Ortmann (Montréal)Fast generation of exchangeable sequences of clusters datahttps://zbmath.org/1536.620132024-07-17T13:47:05.169476Z"Levin, Keith"https://zbmath.org/authors/?q=ai:levin.keith-d"Betancourt, Brenda"https://zbmath.org/authors/?q=ai:betancourt.brendaSummary: Recent advances in Bayesian models for random partitions have led to the formulation and exploration of Exchangeable Sequences of Clusters (ESC) models. Under ESC models, it is the cluster sizes that are exchangeable, rather than the observations themselves. This property is particularly useful for obtaining microclustering behavior, whereby cluster sizes grow sublinearly in the number of observations, as is common in applications such as record linkage, sparse networks and genomics. Unfortunately, the exchangeable clusters property comes at the cost of projectivity. As a consequence, in contrast to more traditional Dirichlet Process or Pitman-Yor process mixture models, samples a priori from ESC models cannot be easily obtained in a sequential fashion and instead require the use of rejection or importance sampling. In this work, drawing on connections between ESC models and discrete renewal theory, we obtain closed-form expressions for certain ESC models and develop faster methods for generating samples a priori from these models compared with the existing state of the art. In the process, we establish analytical expressions for the distribution of the number of clusters under ESC models, which was unknown prior to this work.Explicit and combined estimators for parameters of stable distributionshttps://zbmath.org/1536.620392024-07-17T13:47:05.169476Z"Lévy Véhel, J."https://zbmath.org/authors/?q=ai:levy-vehel.jacques"Philippe, A."https://zbmath.org/authors/?q=ai:philippe.anne"Robet, C."https://zbmath.org/authors/?q=ai:robet.carolineSummary: This article focuses on the estimation of the stability index and scale parameter of stable random variables. We study an estimator based on log-moments,. The main advantage of this estimator is that it has a simple closed-form expression. This allows us to prove an almost sure convergence result as well as a central limit theorem. We show how to improve the accuracy of this estimator by combining it with previously defined ones. The closed-form also enables us to consider the case of non-identically distributed data, and we show that our results still hold provided deviations from stationarity are `small'. Using a centro-symmetrization, we expand the previous estimators to skewed stable variables and we construct a test to check the skewness of the data. As applications, we show numerically that the stability index of multistable Lévy motion may be estimated accurately and consider a financial log\&.A class of dependent Dirichlet processes via latent multinomial processeshttps://zbmath.org/1536.620462024-07-17T13:47:05.169476Z"Nieto-Barajas, Luis E."https://zbmath.org/authors/?q=ai:nieto-barajas.luis-eSummary: We describe a procedure to introduce general dependence structures on a set of Dirichlet processes. Dependence can be in one direction to define a time series or in two directions to define spatial dependencies. More directions can also be considered. Dependence is induced via a set of latent processes and exploit the conjugacy property between the Dirichlet and the multinomial processes to ensure that the marginal law for each element of the set is a Dirichlet process. Dependence is characterized through the correlation between any two elements. Posterior distributions are obtained when we use the set of Dirichlet processes as prior distributions in a Bayesian nonparametric context. Posterior predictive distributions induce partially exchangeable sequences defined by generalized Pólya urns. A numerical example to illustrate is also included.A threshold modeling for nonlinear time series of counts: application to COVID-19 datahttps://zbmath.org/1536.620862024-07-17T13:47:05.169476Z"Shamma, Nisreen"https://zbmath.org/authors/?q=ai:shamma.nisreen"Mohammadpour, Mehrnaz"https://zbmath.org/authors/?q=ai:mohammadpour.mehrnaz"Shirozhan, Masoumeh"https://zbmath.org/authors/?q=ai:shirozhan.masoumehSummary: This article studies a threshold autoregressive model with the dependent thinning structure for modeling nonlinear time series of counts. Some properties are derived for the model and two approaches in estimation are applied, the modified conditional least square and conditional maximum likelihood methods which are adjusted by the Min-Min algorithm. The unknown threshold parameter is estimated using the nested sub-sample search algorithm and the minimum of maximized log-likelihood function methods. The efficiency of the estimators is evaluated using a simulation study. The application of the model is discussed on the COVID-19 data set.A determinantal point process for column subset selectionhttps://zbmath.org/1536.680292024-07-17T13:47:05.169476Z"Belhadji, Ayoub"https://zbmath.org/authors/?q=ai:belhadji.ayoub"Bardenet, Rémi"https://zbmath.org/authors/?q=ai:bardenet.remi"Chainais, Pierre"https://zbmath.org/authors/?q=ai:chainais.pierreSummary: Two popular approaches to dimensionality reduction are principal component analysis, which projects onto a small number of well-chosen but non-interpretable directions, and feature selection, which selects a small number of the original features. Feature selection can be abstracted as selecting the subset of columns of a matrix \(\mathbf{X} \in \mathbb{R}^{N \times d}\) which minimize the approximation error, i.e., the norm of the residual after projecting \(\mathbf{X}\) onto the space spanned by the selected columns. Such a combinatorial optimization is usually impractical, and there has been interest in polynomial-cost, random subset selection algorithms that favour small values of this approximation error. We propose sampling from a projection determinantal point process, a repulsive distribution over column indices that favours diversity among the selected columns. We bound the ratio of the expected approximation error over the optimal error of PCA. These bounds improve over the state-of-the-art bounds of volume sampling when some realistic structural assumptions are satisfied for \(\mathbf{X}\). Numerical experiments suggest that our bounds are tight, and that our algorithms have comparable performance with the double phase algorithm, often considered the practical state-of-the-art.On the capacity of a quantum perceptron for storing biased patternshttps://zbmath.org/1536.810042024-07-17T13:47:05.169476Z"Benatti, Fabio"https://zbmath.org/authors/?q=ai:benatti.fabio"Gramegna, Giovanni"https://zbmath.org/authors/?q=ai:gramegna.giovanni"Mancini, Stefano"https://zbmath.org/authors/?q=ai:mancini.stefano"Nwemadji, Gibbs"https://zbmath.org/authors/?q=ai:nwemadji.gibbsSummary: Although different architectures of quantum perceptrons have been recently put forward, the capabilities of such quantum devices versus their classical counterparts remain debated. Here, we consider random patterns and targets independently distributed with biased probabilities and investigate the storage capacity of a continuous quantum perceptron model that admits a classical limit, thus facilitating the comparison of performances. Such a more general context extends a previous study of the quantum storage capacity where using statistical mechanics techniques in the limit of a large number of inputs, it was proved that no quantum advantages are to be expected concerning the storage properties. This outcome is due to the fuzziness inevitably introduced by the intrinsic stochasticity of quantum devices. We strengthen such an indication by showing that the possibility of indefinitely enhancing the storage capacity for highly correlated patterns, as it occurs in a classical setting, is instead prevented at the quantum level.
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Law of large numbers for the maximum of the two-dimensional Coulomb gas potentialhttps://zbmath.org/1536.820032024-07-17T13:47:05.169476Z"Lambert, Gaultier"https://zbmath.org/authors/?q=ai:lambert.gaultier"Leblé, Thomas"https://zbmath.org/authors/?q=ai:leble.thomas"Zeitouni, Ofer"https://zbmath.org/authors/?q=ai:zeitouni.oferSummary: We derive the leading order asymptotics of the logarithmic potential of a two dimensional Coulomb gas at arbitrary positive temperature. The proof is based on precise evaluation of exponential moments, and the theory of Gaussian multiplicative chaos.Complexity of bipartite spherical spin glasseshttps://zbmath.org/1536.820142024-07-17T13:47:05.169476Z"McKenna, Benjamin"https://zbmath.org/authors/?q=ai:mckenna.benjaminSummary: This paper characterizes the annealed complexity of bipartite spherical spin glasses, both pure and mixed. This means we give exact variational formulas for the asymptotics of the expected numbers of critical points and of local minima. This problem was initially considered by \textit{A. Auffinger} and \textit{W.-K. Chen} [J. Stat. Phys. 157, No. 1, 40--59 (2014; Zbl 1302.82113)], who gave upper and lower bounds on this complexity. We find two surprising connections between pure bipartite and pure single-species spin glasses, which were studied by \textit{A. Auffinger} et al. [Commun. Pure Appl. Math. 66, No. 2, 165--201 (2013; Zbl 1269.82066)]. First, the local minima of any pure bipartite model lie primarily in a low-energy band, similar to the single-species case. Second, for a more restricted set of pure \((p,q)\) bipartite models, the complexity matches exactly that of a pure \(p+q\) single-species model.Optimal activation of halting multi-armed bandit modelshttps://zbmath.org/1536.900802024-07-17T13:47:05.169476Z"Cowan, Wesley"https://zbmath.org/authors/?q=ai:cowan.wesley"Katehakis, Michael N."https://zbmath.org/authors/?q=ai:katehakis.michael-n"Ross, Sheldon M."https://zbmath.org/authors/?q=ai:ross.sheldon-markSummary: We study new types of dynamic allocation problems the \textit{Halting Bandit} models. As an application, we obtain new proofs for the classic Gittins index decomposition result compare [\textit{J. C. Gittins}, J. R. Stat. Soc., Ser. B 41, 148--177 (1979; Zbl 0411.62055)], and recent results of the authors in [\textit{W. Cowan} and \textit{M. N. Katehakis}, Probab. Eng. Inf. Sci. 29, No. 1, 51--76 (2015; Zbl 1414.91104)].
{\copyright} 2023 The Authors. \textit{Naval Research Logistics} published by Wiley Periodicals LLC.On computing medians of marked point process data under edit distancehttps://zbmath.org/1536.901242024-07-17T13:47:05.169476Z"Sukegawa, Noriyoshi"https://zbmath.org/authors/?q=ai:sukegawa.noriyoshi"Suzuki, Shohei"https://zbmath.org/authors/?q=ai:suzuki.shohei"Ikebe, Yoshiko"https://zbmath.org/authors/?q=ai:ikebe.yoshiko-t"Hirata, Yoshito"https://zbmath.org/authors/?q=ai:hirata.yoshitoSummary: In this paper, we consider the problem of computing a median of marked point process data under an edit distance. We formulate this problem as a binary linear program, and propose to solve it to optimality by software. We show results of numerical experiments to demonstrate the effectiveness of the proposed method and its application in earthquake prediction.Utility maximization of the exponential Lévy switching modelshttps://zbmath.org/1536.911622024-07-17T13:47:05.169476Z"Dong, Y."https://zbmath.org/authors/?q=ai:dong.yuchao"Vostrikova, L."https://zbmath.org/authors/?q=ai:vostrikova.lioudmilaSummary: This article is devoted to maximization of HARA (hyperbolic absolute risk aversion) utilities of the exponential Lévy switching processes on a finite time interval via the dual method. The description of all \(f\)-divergence minimal martingale measures and the expression of their Radon-Nikodým densities involving the Hellinger and Kulback-Leibler processes are given. The optimal strategies in progressively enlarged filtration for the maximization of HARA utilities as well as the values of the corresponding maximal expected utilities are derived. As an example, the Brownian switching model is presented with financial interpretations of the results via the value process.Valuation of variable annuities with guaranteed minimum maturity benefits and periodic feeshttps://zbmath.org/1536.912742024-07-17T13:47:05.169476Z"Ai, Meiqiao"https://zbmath.org/authors/?q=ai:ai.meiqiao"Wang, Yunyun"https://zbmath.org/authors/?q=ai:wang.yunyun"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhimin.1"Zhu, Dan"https://zbmath.org/authors/?q=ai:zhu.danSummary: This paper focuses on the valuation of variable annuities with a guaranteed minimum maturity benefit under a regime-switching Lévy model. The model allows policyholders to surrender their annuities and receive a surrender benefit at predetermined tenor times before maturity. Additionally, we consider a state-dependent periodic fee structure where fees are deducted from the policyholder's account if it exceeds a certain level at discrete time points. Incorporating this fee structure, the Fourier cosine series expansion method based on characteristic functions is employed to determine the values and optimal surrender strategies for variable annuity contracts. Finally, we provide a comprehensive set of numerical examples to demonstrate and assess the effectiveness of our approach thoroughly.Optimal reinsurance design under solvency constraintshttps://zbmath.org/1536.912752024-07-17T13:47:05.169476Z"Avanzi, Benjamin"https://zbmath.org/authors/?q=ai:avanzi.benjamin"Lau, Hayden"https://zbmath.org/authors/?q=ai:lau.hayden"Steffensen, Mogens"https://zbmath.org/authors/?q=ai:steffensen.mogensSummary: We consider the optimal risk transfer from an insurance company to a reinsurer. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some crucial distinctions. In particular, the insurance company's surplus is here (as is routinely the case) approximated by a Brownian motion, as opposed to the geometric Brownian motion used to model assets in finance. Furthermore, risk exposure is dialled `down' via reinsurance, rather than `up' via risky investments. This leads to interesting qualitative differences in the optimal designs. In this paper, using the martingale method, we derive the optimal design as a function of proportional, non-cheap reinsurance design that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider several realistic constraints on the terminal value: a strict lower boundary, the probability (value at risk) constraint, and the expected shortfall (conditional value at risk) constraints under the \(\mathbb{P}\) and \(\mathbb{Q}\) measures, respectively. In all cases, the optimal reinsurance designs boil down to a combination of proportional protection and option-like protection (stop-loss) of the residual proportion with various deductibles. Proportions and deductibles are set such that the initial capital is fully allocated. Comparison of the optimal designs with the optimal portfolios in finance is particularly interesting. Results are illustrated.Price impact without averaginghttps://zbmath.org/1536.913092024-07-17T13:47:05.169476Z"Bellani, Claudio"https://zbmath.org/authors/?q=ai:bellani.claudio"Brigo, Damiano"https://zbmath.org/authors/?q=ai:brigo.damiano"Pakkanen, Mikko S."https://zbmath.org/authors/?q=ai:pakkanen.mikko-s"Sánchez-Betancourt, Leandro"https://zbmath.org/authors/?q=ai:sanchez-betancourt.leandroSummary: We present a method to estimate price impact in order-driven markets that does not require averaging over executions or scenarios. Given order book data associated with one single execution of a sell metaorder, we estimate its contribution to price decrease during the trade. We do so by modelling the limit order book using a state-dependent Hawkes process, and by defining the price impact profile of the execution as a function of the compensator of the state-dependent Hawkes process. We apply our method to a dataset from NASDAQ, and we conclude that the scheduling of sell child orders has a bigger impact on price than their sizes.Exogenous and endogenous price jumps belong to different dynamical classeshttps://zbmath.org/1536.913212024-07-17T13:47:05.169476Z"Marcaccioli, Riccardo"https://zbmath.org/authors/?q=ai:marcaccioli.riccardo"Bouchaud, Jean-Philippe"https://zbmath.org/authors/?q=ai:bouchaud.jean-philippe"Benzaquen, Michael"https://zbmath.org/authors/?q=ai:benzaquen.michaelSummary: Synchronising a database of stock specific news with 5 years worth of order book data on 300 stocks, we show that abnormal price movements following news releases (exogenous) exhibit markedly different dynamical features from those arising spontaneously (endogenous). On average, large volatility fluctuations induced by exogenous events occur abruptly and are followed by a decaying power-law relaxation, while endogenous price jumps are characterized by progressively accelerating growth of volatility, also followed by a power-law relaxation, but slower than for exogenous jumps. Remarkably, our results are reminiscent of what is observed in different contexts, namely Amazon book sales and YouTube views. Finally, we show that fitting power-laws to \textit{individual} volatility profiles allows one to classify large events into endogenous and exogenous dynamical classes, without relying on the news feed.Mild to classical solutions for XVA equations under stochastic volatilityhttps://zbmath.org/1536.913292024-07-17T13:47:05.169476Z"Brigo, Damiano"https://zbmath.org/authors/?q=ai:brigo.damiano"Graceffa, Federico"https://zbmath.org/authors/?q=ai:graceffa.federico"Kalinin, Alexander"https://zbmath.org/authors/?q=ai:kalinin.alexander-v|kalinin.alexandrSummary: We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.Wiener spiral for volatility modelinghttps://zbmath.org/1536.913312024-07-17T13:47:05.169476Z"Fukasawa, M."https://zbmath.org/authors/?q=ai:fukasawa.masaakiThis paper provides an elementary introduction to the class of ``rough volatility models'', that have gained a lot of popularity in the mathematical finance literature. These models are driven by fractional Brownian motions (or ``Wiener spirals'' in the terminology of Kolmogorov), a class of processes which allows to generate both long range dependence and paths even rougher than standard Brownian motion. The first part of the paper recalls some basic results about these processes and their properties, before introducing the typical models used in a financial context. The two main parts of the paper then look at two distinct sets of empirical stylized facts that can be reproduced in a parsimonious manner by rough volatility models. To wit, the author first discusses how these models can fit the term structure of volatilities implied by the prices of traded option contracts. Subsequently, the author turns to statistical properties of time series of asset prices and shows that the rough volatility models also produce correlations between prices and their (realized) volatilities consistent with empirical data.
Reviewer: Johannes Muhle-Karbe (London)Approximate pricing of derivatives under fractional stochastic volatility modelhttps://zbmath.org/1536.913332024-07-17T13:47:05.169476Z"Han, Y."https://zbmath.org/authors/?q=ai:han.yuecai"Zheng, X."https://zbmath.org/authors/?q=ai:zheng.xudong.3Summary: This paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein-Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein-Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.Model-free weak no-arbitrage and superhedging under transaction costs beyond efficient frictionhttps://zbmath.org/1536.913362024-07-17T13:47:05.169476Z"Ryom, Songchol"https://zbmath.org/authors/?q=ai:ryom.songchol"Ri, Inchol"https://zbmath.org/authors/?q=ai:ri.incholSummary: The paper is continuation of \textit{M.-C. Kim} and \textit{S.-C. Ryom} [Math. Financ. Econ. 16, No. 4, 713--747 (2022; Zbl 1498.91449)], in which a pathwise superhedging duality was proved for multidimensional contingent claims under model-free strict no-arbitrage and efficient frictions. We consider a two-dimensional market with transaction costs beyond efficient friction in a model-free framework. We get a condition to hold model-free weak no-arbitrage and prove a superhedging duality under model-free weak no-arbitrage.Monte Carlo simulation for trading under a Lévy-driven mean-reverting frameworkhttps://zbmath.org/1536.913482024-07-17T13:47:05.169476Z"Leung, Tim"https://zbmath.org/authors/?q=ai:leung.tim"Lu, Kevin W."https://zbmath.org/authors/?q=ai:lu.kevin-wSummary: We present a Monte Carlo approach to pairs trading on mean-reverting spreads modelled by Lévy-driven Ornstein-Uhlenbeck processes. Specifically, we focus on using a variance gamma driving process, an infinite activity pure jump process to allow for more flexible models of the price spread than is available in the classical model. However, this generalization comes at the cost of not having analytic formulas, so we apply Monte Carlo methods to determine optimal trading levels and develop a variance reduction technique using control variates. Within this framework, we numerically examine how the optimal trading strategies are affected by the parameters of the model. In addition, we extend our method to bivariate spreads modelled using a weak variance alpha-gamma driving process, and explore the effect of correlation on these trades.Projection and contraction method for the valuation of American options under regime switchinghttps://zbmath.org/1536.913502024-07-17T13:47:05.169476Z"Song, Haiming"https://zbmath.org/authors/?q=ai:song.haiming"Xu, Jingbo"https://zbmath.org/authors/?q=ai:xu.jingbo"Yang, Jinda"https://zbmath.org/authors/?q=ai:yang.jinda"Li, Yutian"https://zbmath.org/authors/?q=ai:li.yutianSummary: This paper proposes an efficient numerical algorithm to evaluate American put options under regime switching. A set of variational inequalities could describe the pricing model, which is equivalent to a coupled parabolic free boundary problem. With variable substitutions and truncation techniques, the original pricing problem is transformed into a coupled linear complementarity problem on a bounded rectangular domain. Then, the variational problem related to the linear complementary problem is obtained. Furthermore, a finite difference method in the temporal direction and a finite element method in the spatial direction lead to a full-discretized approximation of the variational problem. Based on the positive definiteness of the discretized matrix, a projection and contraction method is adopted for the resulting discretized variational problem. Finally, several numerical simulations are carried out to illustrate the efficiency of the proposed method compared with existing methods.Ultrametric diffusion equation on energy landscape to model disease spread in hierarchic socially clustered populationhttps://zbmath.org/1536.921322024-07-17T13:47:05.169476Z"Khrennikov, Andrei"https://zbmath.org/authors/?q=ai:khrennikov.andrei-yuThe article is devoted to modeling of a COVID-19 disease spread approaching a herd immunity. Graphs with \(p\)-adic metric are used. Stochastic processes on them are analysed. A social clustering is utilized with energy barriers. The known results on stochastic processes in spaces over \(\mathbf{Q}_p\) are applied. Other results on stochastic processes on manifolds over fields supplied with multiplicative norms satisfying the strong triangle inequality also can be helpful for this, for example, [\textit{S. V. Ludkovsky}, Int. J. Math. Math. Sci. 2004, No. 29--32, 1633--1651 (2004; Zbl 1094.60022)].
Reviewer: Sergey Ludkovsky (Moskva)Inferring stochastic group interactions within structured populations via coupled autoregressionhttps://zbmath.org/1536.921662024-07-17T13:47:05.169476Z"McGrane-Corrigan, Blake"https://zbmath.org/authors/?q=ai:mcgrane-corrigan.blake"Mason, Oliver"https://zbmath.org/authors/?q=ai:mason.oliver"de Andrade Moral, Rafael"https://zbmath.org/authors/?q=ai:de-andrade-moral.rafaelSummary: The internal behaviour of a population is an important feature to take account of when modelling its dynamics. In line with kin selection theory, many social species tend to cluster into distinct groups in order to enhance their overall population fitness. Temporal interactions between populations are often modelled using classical mathematical models, but these sometimes fail to delve deeper into the, often uncertain, relationships within populations. Here, we introduce a stochastic framework that aims to capture the interactions of animal groups and an auxiliary population over time. We demonstrate the model's capabilities, from a Bayesian perspective, through simulation studies and by fitting it to predator-prey count time series data. We then derive an approximation to the group correlation structure within such a population, while also taking account of the effect of the auxiliary population. We finally discuss how this approximation can lead to ecologically realistic interpretations in a predator-prey context. This approximation also serves as verification to whether the population in question satisfies our various assumptions. Our modelling approach will be useful for empiricists for monitoring groups within a conservation framework and also theoreticians wanting to quantify interactions, to study cooperation and other phenomena within social populations.Local intraspecific aggregation in phytoplankton model communities: spatial scales of occurrence and implications for coexistencehttps://zbmath.org/1536.921672024-07-17T13:47:05.169476Z"Picoche, Coralie"https://zbmath.org/authors/?q=ai:picoche.coralie"Young, William R."https://zbmath.org/authors/?q=ai:young.william-r"Barraquand, Frédéric"https://zbmath.org/authors/?q=ai:barraquand.fredericSummary: The coexistence of multiple phytoplankton species despite their reliance on similar resources is often explained with mean-field models assuming mixed populations. In reality, observations of phytoplankton indicate spatial aggregation at all scales, including at the scale of a few individuals. Local spatial aggregation can hinder competitive exclusion since individuals then interact mostly with other individuals of their own species, rather than competitors from different species. To evaluate how microscale spatial aggregation might explain phytoplankton diversity maintenance, an individual-based, multispecies representation of cells in a hydrodynamic environment is required. We formulate a three-dimensional and multispecies individual-based model of phytoplankton population dynamics at the Kolmogorov scale. The model is studied through both simulations and the derivation of spatial moment equations, in connection with point process theory. The spatial moment equations show a good match between theory and simulations. We parameterized the model based on phytoplankters' ecological and physical characteristics, for both large and small phytoplankton. Defining a zone of potential interactions as the overlap between nutrient depletion volumes, we show that local species composition -- within the range of possible interactions -- depends on the size class of phytoplankton. In small phytoplankton, individuals remain in mostly monospecific clusters. Spatial structure therefore favours intra- over inter-specific interactions for small phytoplankton, contributing to coexistence. Large phytoplankton cell neighbourhoods appear more mixed. Although some small-scale self-organizing spatial structure remains and could influence coexistence mechanisms, other factors may need to be explored to explain diversity maintenance in large phytoplankton.The role of memory-based movements in the formation of animal home rangeshttps://zbmath.org/1536.921762024-07-17T13:47:05.169476Z"Ranc, Nathan"https://zbmath.org/authors/?q=ai:ranc.nathan"Cain, John W."https://zbmath.org/authors/?q=ai:cain.john-w"Cagnacci, Francesca"https://zbmath.org/authors/?q=ai:cagnacci.francesca"Moorcroft, Paul R."https://zbmath.org/authors/?q=ai:moorcroft.paul-rSummary: Most animals live in spatially-constrained home ranges. The prevalence of this space-use pattern in nature suggests that general biological mechanisms are likely to be responsible for their occurrence. Individual-based models of animal movement in both theoretical and empirical settings have demonstrated that the revisitation of familiar areas through memory can lead to the formation of stable home ranges. Here, we formulate a deterministic, mechanistic home range model that includes the interplay between a bi-component memory and resource preference, and evaluate resulting patterns of space-use. We show that a bi-component memory process can lead to the formation of stable home ranges and control its size, with greater spatial memory capabilities being associated with larger home range size. The interplay between memory and resource preferences gives rise to a continuum of space-use patterns-from spatially-restricted movements into a home range that is influenced by local resource heterogeneity, to diffusive-like movements dependent on larger-scale resource distributions, such as in nomadism. Future work could take advantage of this model formulation to evaluate the role of memory in shaping individual performance in response to varying spatio-temporal resource patterns.A comparison of three algorithms in the filtering of a Markov-modulated non-homogeneous Poisson processhttps://zbmath.org/1536.939272024-07-17T13:47:05.169476Z"Li, Yuying"https://zbmath.org/authors/?q=ai:li.yuying"Mamon, Rogemar"https://zbmath.org/authors/?q=ai:mamon.rogemar-sSummary: A Markov-modulated non-homogeneous Poisson process (MMNPP), whose intensity process is designed to capture both the cyclical and nonrecurring trends, is considered for modelling the total count of cyber incidents. Extending the expectation-maximisation (EM) algorithm for the current MMPP literature, we derive the filters and smoothers to support the MMNPP online parameter estimation. A scaling transformation is introduced to address the numerical issue for large data sizes whilst maintaining accuracy. The filter- and smoother-based EM algorithms are then benchmarked to the maximum likelihood-based EM algorithm at the theoretical level. The differences emerge in the E-step of the EM procedure. Both the filtering and smoothing schemes, in conjunction with the change-of-measure technique, avoid the computing complication caused by the hidden regimes. In contrast to the usual EM algorithm, the said two algorithms could be implemented given only the incident counts data without the specific times of jumps. Within the data compiled by the U.S. Department of Health and Human Services, the filter-based algorithm performs better than the algorithm involving smoothers. The benchmarked algorithm may do well in calibration under the presence of extreme incident counts with an extremely low frequency; however, overfitting may occur. For most practical applications involving 2 or 3 regimes, both algorithms are superior when it comes to efficiency, real-time update, and low computational cost. The benchmarked algorithm is better when there are more regimes under relatively closer intensities. Overall, the filter-based algorithm gives better estimation, especially if there is a low-frequency regime and the flexible binning of the data set is an important consideration.Stochastic filtering under model ambiguityhttps://zbmath.org/1536.939462024-07-17T13:47:05.169476Z"Zhang, Jiaqi"https://zbmath.org/authors/?q=ai:zhang.jiaqi"Xiong, Jie"https://zbmath.org/authors/?q=ai:xiong.jie|xiong.jie.1The filtering (estimation) problem for systems described by stochastic differential equations has been studied extensively since the principal works of \textit{R. L. Stratonovich} [Teor. Veroyatn. Primen. 5, 172--195 (1960; Zbl 0094.13101)] and \textit{H. J. Kushner} [J. Differ. Equations 3, 179--190 (1967; Zbl 0158.16801)]. The optimal filtering equation is a non-linear stochastic partial differential equation (SPDE), which is usually called the Kushner--Stratonovich equation or the Kushner--FKK equation. The developed classical filtering (estimation) methods are not directly applicable in practice since the exact spectral structure of both the signal and observation processes is not usually available. Therefore, it is reasonable to consider the estimates, called minimax-robust, which minimize the maximum of the errors for all spectral measures from a given set of admissible spectral measures simultaneously. The authors of this article investigate the filtering problem under model uncertainty for the model with real-valued signal and observation processes described by the equations
\[ \begin{cases}
dX_t=b(X_t)dt+\sigma(X_t)dW_t,\quad &X_0=x,\\
dY_t=h(X_t)dt+dB_t,\quad &Y_0=0,
\end{cases} \]
where the coefficients \(b,\sigma,h\) are continuous real functions, \((W_t,B_t)\) is a 2-dimensional standard Brownian motion under probability measure \(P\in\mathcal P\). The probability measure \(P\) serves as an evaluation criterion for the signal process observed by external observers. The probability measure set \(\mathcal P\) is considered to encompass all evaluation criteria with the ambiguity parameters \(\theta\in\Theta\). The estimate of the signal process is obtained by minimizing the squared error in the worst-case scenario \[\min_{\xi}\max_{P\in\mathcal P} \mathbb E^P[|X_t-\xi|^2],\] where \(\mathbb E^P\) is the expectation with respect to the probability measure \(P\), and \(\xi\) is over all \(\mathcal G\)-measurable random variables. Here \(\mathcal{G}\equiv\sigma (Y_s:s\leq t)\). The authors start with proving the existence of the optimal control that minimizes the error under the most unfavorable evaluation criteria, relying on partially observable information. Then, by utilizing the mini-max theorem, they interchanged the order of extremum problems and characterized the optimal control. The most favorable evaluation criteria, namely, the optimal probability measure is described.
For more related results and references see [\textit{S. A. Kassam} and \textit{H. V. Poor}, Proc. IEEE 73, No. 3, 433--481 (1985; Zbl 0569.62084); \textit{M. Moklyachuk}, ``Minimax-robust estimation problems for stationary stochastic sequences'', Stat. Optim. Inf. Comput. 3, No. 4, 348--419 (2015); \textit{M. Luz} and \textit{M. Moklyachuk}, Estimation of stochastic processes with stationary increments and cointegrated sequences. London: ISTE; Hoboken, NJ: John Wiley \& Sons (2019; Zbl 1430.62007)].
Reviewer: Mikhail P. Moklyachuk (Kyïv)Optimal transport between Gaussian stationary processeshttps://zbmath.org/1536.939962024-07-17T13:47:05.169476Z"Zorzi, Mattia"https://zbmath.org/authors/?q=ai:zorzi.mattiaEditorial remark: No review copy delivered.