Stochastic Processes and their Applications最新文献_第8页

Wasserstein convergence rates for empirical measures of random subsequence of {nα} {nα}随机子序列经验测量的瓦瑟斯坦收敛率

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-23 DOI: 10.1016/j.spa.2024.104534

Bingyao Wu , Jie-Xiang Zhu

{"title":"Wasserstein convergence rates for empirical measures of random subsequence of {nα}","authors":"Bingyao Wu , Jie-Xiang Zhu","doi":"10.1016/j.spa.2024.104534","DOIUrl":"10.1016/j.spa.2024.104534","url":null,"abstract":"<div><div>Fix an irrational number <span><math><mi>α</mi></math></span>. Let <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></mrow></math></span> be independent, identically distributed, integer-valued random variables with characteristic function <span><math><mi>φ</mi></math></span>, and let <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span> be the partial sums. Consider the random walk <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>α</mi><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> on the torus, where <span><math><mrow><mo>{</mo><mi>⋅</mi><mo>}</mo></mrow></math></span> denotes the fractional part. We study the long time asymptotic behavior of the empirical measure of this random walk to the uniform distribution under the general <span><math><mi>p</mi></math></span>-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of <span><math><mi>α</mi></math></span> and the Hölder continuity of the characteristic function <span><math><mi>φ</mi></math></span> at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in Ambrosio et al. (2019) and the continued fraction representation of the irrational number <span><math><mi>α</mi></math></span>.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104534"},"PeriodicalIF":1.1,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722995","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}

引用次数: 0

Diffusive limit approximation of pure jump optimal ergodic control problems

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-22 DOI: 10.1016/j.spa.2024.104536

Marc Abeille , Bruno Bouchard , Lorenzo Croissant

{"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}

引用次数: 0

Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation 非负保留卷积核。应用于闭凸域中的随机伏特拉方程及其近似方法

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-22 DOI: 10.1016/j.spa.2024.104535

Aurélien Alfonsi

引用次数: 0

Correlation structure and resonant pairs for arithmetic random waves 算术随机波的相关结构和共振对

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-20 DOI: 10.1016/j.spa.2024.104525

Valentina Cammarota , Riccardo W. Maffucci , Domenico Marinucci , Maurizia Rossi

引用次数: 0

Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus 通过马利亚文微积分对多尺度扩散系统进行定量波动分析

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-14 DOI: 10.1016/j.spa.2024.104524

S. Bourguin , K. Spiliopoulos

引用次数: 0

Model selection for Markov random fields on graphs under a mixing condition 混合条件下图上马尔可夫随机场的模型选择

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-14 DOI: 10.1016/j.spa.2024.104523

Florencia Leonardi , Magno T.F. Severino

引用次数: 0

An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics 带权重的遍历定理及其在随机测量、均质化和流体力学中的应用

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-13 DOI: 10.1016/j.spa.2024.104522

Alessandra Faggionato

{"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}

引用次数: 0

The isoperimetric problem for convex hulls and large deviations rate functionals of random walks 凸壳的等周问题和随机游走的大偏差率函数

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-05 DOI: 10.1016/j.spa.2024.104519

Vladislav Vysotsky

{"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}

引用次数: 0

The site frequency spectrum for coalescing Brownian motion 凝聚布朗运动的场地频谱

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-04 DOI: 10.1016/j.spa.2024.104521

Yubo Shuai

引用次数: 0

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line 正半线上弹性杀伤扩散系统的带罗宾型边界的 SPDE

IF 1.1 2区数学

Stochastic Processes and their Applications Pub Date : 2024-11-02 DOI: 10.1016/j.spa.2024.104520

Ben Hambly , Julian Meier , Andreas Søjmark

引用次数: 0