{"title":"Diffusive limit approximation of pure jump optimal ergodic control problems","authors":"Marc Abeille , Bruno Bouchard , Lorenzo Croissant","doi":"10.1016/j.spa.2024.104536","DOIUrl":"10.1016/j.spa.2024.104536","url":null,"abstract":"<div><div>Motivated by the design of fast reinforcement learning algorithms, see (Croissant et al., 2024), we study the diffusive limit of a class of pure jump ergodic stochastic control problems. We show that, whenever the intensity of jumps <span><math><msup><mrow><mi>ɛ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> is large enough, the approximation error is governed by the Hölder regularity of the Hessian matrix of the solution to the limit ergodic partial differential equation and is, indeed, of order <span><math><msup><mrow><mi>ɛ</mi></mrow><mrow><mfrac><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> for all <span><math><mrow><mi>γ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. This extends to this context the results of Abeille et al. (2023) obtained for finite horizon problems. Using the limit as an approximation, instead of directly solving the pre-limit problem, allows for a very significant reduction in the numerical resolution cost of the control problem. Additionally, we explain how error correction terms of this approximation can be constructed under appropriate smoothness assumptions. Finally, we quantify the error induced by the use of the Markov control policy constructed from the numerical finite difference scheme associated to the limit diffusive problem, which seems to be new in the literature and of independent interest.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104536"},"PeriodicalIF":1.1,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation","authors":"Aurélien Alfonsi","doi":"10.1016/j.spa.2024.104535","DOIUrl":"10.1016/j.spa.2024.104535","url":null,"abstract":"<div><div>This work defines and studies one-dimensional convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of Hawkes processes, this property allows to get the nonnegativity of the integral. We give characterizations of these kernels and show in particular that completely monotone kernels preserve nonnegativity. We then apply these results to analyze the stochastic invariance of a closed convex set by Stochastic Volterra Equations. We also get a comparison result in dimension one. Last, when the kernel is a positive linear combination of decaying exponential functions, we present a second order approximation scheme for the weak error that stays in the closed convex domain under suitable assumptions. We apply these results to the rough Heston model and give numerical illustrations.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104535"},"PeriodicalIF":1.1,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correlation structure and resonant pairs for arithmetic random waves","authors":"Valentina Cammarota , Riccardo W. Maffucci , Domenico Marinucci , Maurizia Rossi","doi":"10.1016/j.spa.2024.104525","DOIUrl":"10.1016/j.spa.2024.104525","url":null,"abstract":"<div><div>The geometry of Arithmetic Random Waves has been extensively investigated in the last fifteen years, starting from the seminal papers (Rudnick and Wigman, 2008; Oravecz et al., 2008). In this paper we study the correlation structure among different functionals such as nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of <em>resonant pairs</em> of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on the 2-dimensional case but we discuss some specific extensions to dimension 3.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104525"},"PeriodicalIF":1.1,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus","authors":"S. Bourguin , K. Spiliopoulos","doi":"10.1016/j.spa.2024.104524","DOIUrl":"10.1016/j.spa.2024.104524","url":null,"abstract":"<div><div>We study fluctuations of small noise multiscale diffusions around their homogenized deterministic limit. We derive quantitative rates of convergence of the fluctuation processes to their Gaussian limits in the appropriate Wasserstein metric requiring detailed estimates of the first and second order Malliavin derivative of the slow component. We study a fully coupled system and the derivation of the quantitative rates of convergence depends on a very careful decomposition of the first and second Malliavin derivatives of the slow and fast component to terms that have different rates of convergence depending on the strength of the noise and timescale separation parameter.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"180 ","pages":"Article 104524"},"PeriodicalIF":1.1,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Model selection for Markov random fields on graphs under a mixing condition","authors":"Florencia Leonardi , Magno T.F. Severino","doi":"10.1016/j.spa.2024.104523","DOIUrl":"10.1016/j.spa.2024.104523","url":null,"abstract":"<div><div>We propose a global model selection criterion to estimate the graph of conditional dependencies of a random vector. By global criterion, we mean optimizing a function over the set of possible graphs, eliminating the need to estimate individual neighborhoods and subsequently combine them to estimate the graph. We prove the almost sure convergence of the graph estimator. This convergence holds, provided the data is a realization of a multivariate stochastic process that satisfies a polynomial mixing condition. These are the first results to show the consistency of a model selection criterion for Markov random fields on graphs under non-independent data.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"180 ","pages":"Article 104523"},"PeriodicalIF":1.1,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics","authors":"Alessandra Faggionato","doi":"10.1016/j.spa.2024.104522","DOIUrl":"10.1016/j.spa.2024.104522","url":null,"abstract":"<div><div>We prove a multidimensional ergodic theorem with weighted averages for the action of the group <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> on a probability space. At level <span><math><mi>n</mi></math></span> weights are of the form <span><math><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mi>ψ</mi><mrow><mo>(</mo><mi>j</mi><mo>/</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>j</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, for real functions <span><math><mi>ψ</mi></math></span> decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from [Faggionato 2023, Theorem 4.4]. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in [Faggionato 2022, Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"180 ","pages":"Article 104522"},"PeriodicalIF":1.1,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The isoperimetric problem for convex hulls and large deviations rate functionals of random walks","authors":"Vladislav Vysotsky","doi":"10.1016/j.spa.2024.104519","DOIUrl":"10.1016/j.spa.2024.104519","url":null,"abstract":"<div><div>We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, such a scaled limit trajectory <span><math><mi>h</mi></math></span> solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of <span><math><mi>h</mi></math></span> is replaced by the large deviations rate functional <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mi>I</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>t</mi></mrow></math></span> and <span><math><mi>I</mi></math></span> is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler–Lagrange equation, which we solve explicitly for every <span><math><mi>I</mi></math></span>. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"180 ","pages":"Article 104519"},"PeriodicalIF":1.1,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The site frequency spectrum for coalescing Brownian motion","authors":"Yubo Shuai","doi":"10.1016/j.spa.2024.104521","DOIUrl":"10.1016/j.spa.2024.104521","url":null,"abstract":"<div><div>Motivated by the goal of understanding the genealogy of a sample from an expanding population in the plane, we consider coalescing Brownian motion on the circle. For this model, we establish a weak law of large numbers for the site frequency spectrum. A parallel result holds for a localized version where the genealogy is modeled by coalescing Brownian motion on the line.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"180 ","pages":"Article 104521"},"PeriodicalIF":1.1,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line","authors":"Ben Hambly , Julian Meier , Andreas Søjmark","doi":"10.1016/j.spa.2024.104520","DOIUrl":"10.1016/j.spa.2024.104520","url":null,"abstract":"<div><div>We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line in the limit as the number of particles goes to infinity. We establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles, which is a weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We show that this density process has good <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-regularity properties in the interior of the domain but may exhibit singularities on the boundary at a dense set of times. We make connections to the corresponding absorbing and reflecting SPDEs as the elastic parameter varies.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104520"},"PeriodicalIF":1.1,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142664023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Compound Poisson distributions for random dynamical systems using probabilistic approximations","authors":"Lucas Amorim , Nicolai Haydn , Sandro Vaienti","doi":"10.1016/j.spa.2024.104511","DOIUrl":"10.1016/j.spa.2024.104511","url":null,"abstract":"<div><div>We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with the region where uniformity breaks. Based on annealed and quenched polynomial decay of correlations, our quenched result adopts annealed Kac-type time-normalization and finds limits to be noise-independent. The technique involves a probabilistic block-approximation where the quenched hit-counting function up to annealed Kac-normalized time is split into equally sized blocks which are mimicked by an independency of random variables distributed just like each of them. The theory is made operational due to a result that allows certain hitting quantities to be recovered from return quantities. Our application is to a class of random piecewise expanding one-dimensional systems, casting new light on the well-known deterministic dichotomy between periodic and aperiodic points, their usual extremal index formula <span><math><mrow><mo>EI</mo><mo>=</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>J</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, and recovering the Polya–Aeppli case for general Bernoulli-driven systems, but distinct behavior otherwise.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104511"},"PeriodicalIF":1.1,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}