Stochastic Processes and their Applications最新文献

{"title":"On the limit theory of mean field optimal stopping with non-Markov dynamics and common noise","authors":"Xihao He","doi":"10.1016/j.spa.2025.104681","DOIUrl":"10.1016/j.spa.2025.104681","url":null,"abstract":"<div><div>This paper focuses on a mean-field optimal stopping problem with non-Markov dynamics and common noise, inspired by Talbi et al. (2025,2022). The goal is to establish the limit theory and demonstrate the equivalence of the value functions between weak and strong formulations. The difference between the strong and weak formulations lies in the source of randomness determining the stopping time on a canonical space. In the strong formulation, the randomness of the stopping time originates from Brownian motions. In contrast, this may not necessarily be the case in the weak formulation. Additionally, a <span><math><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></math></span>-Hypothesis-type condition is introduced to guarantee the equivalence of the value functions. The limit theory encompasses the convergence of the value functions and solutions of the large population optimal stopping problem towards those of the mean-field limit, and it shows that every solution of the mean field optimal stopping problem can be approximated by solutions of the large population optimal stopping problem.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104681"},"PeriodicalIF":1.1,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106651","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":"Taming under isoperimetry","authors":"Iosif Lytras ,&nbsp;Sotirios Sabanis","doi":"10.1016/j.spa.2025.104684","DOIUrl":"10.1016/j.spa.2025.104684","url":null,"abstract":"<div><div>In this article we propose a novel taming Langevin-based scheme called <span><math><mi>sTULA</mi></math></span> to sample from distributions with superlinearly growing log-gradient which also satisfy a Log-Sobolev inequality. We derive non-asymptotic convergence bounds in <span><math><mrow><mi>K</mi><mi>L</mi></mrow></math></span> and consequently total variation and Wasserstein-2 distance from the target measure. Non-asymptotic convergence guarantees are provided for the performance of the new algorithm as an optimizer. Finally, some theoretical results on isoperimertic inequalities for distributions with superlinearly growing gradients are provided. Key findings are a Log-Sobolev inequality with constant independent of the dimension, in the presence of a higher order regularization and a Poincaré inequality with constant independent of temperature and dimension under a novel non-convex theoretical framework.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104684"},"PeriodicalIF":1.1,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143950503","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":"Quantitative hydrodynamics for a generalized contact model","authors":"Julian Amorim,&nbsp;Milton Jara,&nbsp;Yangrui Xiang","doi":"10.1016/j.spa.2025.104680","DOIUrl":"10.1016/j.spa.2025.104680","url":null,"abstract":"<div><div>We derive a quantitative version of the hydrodynamic limit obtained in Chariker et al. (2023) for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-speed of convergence of the empirical density of states in a generalized contact process defined over a <span><math><mi>d</mi></math></span>-dimensional torus of size <span><math><mi>n</mi></math></span> is of the optimal order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>. In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104680"},"PeriodicalIF":1.1,"publicationDate":"2025-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143950504","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":"Some quenched and annealed limit theorems for superprocesses in random environments","authors":"Zeteng Fan,&nbsp;Jieliang Hong ,&nbsp;Jie Xiong","doi":"10.1016/j.spa.2025.104686","DOIUrl":"10.1016/j.spa.2025.104686","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>X</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> be a superprocess in a random environment described by a Gaussian noise <span><math><mrow><mi>W</mi><mo>=</mo><mrow><mo>{</mo><mi>W</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>}</mo></mrow></mrow></math></span> white in time and colored in space with correlation kernel <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>. When <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, under the condition that the correlation function <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> is bounded above by some appropriate function <span><math><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>, we present the quenched and annealed Strong Law of Large Numbers and the Central Limit Theorems regarding the weighted occupation measure <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><msub><mrow><mi>X</mi></mrow><mrow><mi>s</mi></mrow></msub><mi>d</mi><mi>s</mi></mrow></math></span> as <span><math><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></math></span>.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104686"},"PeriodicalIF":1.1,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143950502","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":"Averaging principle for semilinear slow–fast rough partial differential equations","authors":"Miaomiao Li ,&nbsp;Yunzhang Li ,&nbsp;Bin Pei ,&nbsp;Yong Xu","doi":"10.1016/j.spa.2025.104683","DOIUrl":"10.1016/j.spa.2025.104683","url":null,"abstract":"<div><div>In this paper, we investigate the averaging principle for a class of semilinear slow–fast partial differential equations driven by finite-dimensional rough multiplicative noise. Specifically, the slow component is driven by a general random <span><math><mi>γ</mi></math></span>-Hölder rough path for some <span><math><mrow><mi>γ</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, while the fast component is driven by an Itô-type Brownian rough path. Using controlled rough path theory and the classical Khasminskii’s time discretization scheme, we demonstrate that the slow component converges strongly to the solution of the corresponding averaged equation under the Hölder topology.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104683"},"PeriodicalIF":1.1,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084498","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":"Stationary entrance chains and applications to random walks","authors":"Aleksandar Mijatović ,&nbsp;Vladislav Vysotsky","doi":"10.1016/j.spa.2025.104668","DOIUrl":"10.1016/j.spa.2025.104668","url":null,"abstract":"<div><div>For a Markov chain <span><math><mi>Y</mi></math></span> with values in a Polish space, consider the <em>entrance chain</em> obtained by sampling <span><math><mi>Y</mi></math></span> at the moments when it enters a fixed set <span><math><mi>A</mi></math></span> from its complement <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>. Similarly, consider the <em>exit chain</em>, obtained by sampling <span><math><mi>Y</mi></math></span> at the exit times from <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> to <span><math><mi>A</mi></math></span>. We use the method of inducing from ergodic theory to study invariant measures of these two types of Markov chains in the case when the initial chain <span><math><mi>Y</mi></math></span> has a known invariant measure. We give explicit formulas for invariant measures of the entrance and exit chains under certain recurrence-type assumptions on <span><math><mi>A</mi></math></span> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>, which apply even for transient chains. Then we study uniqueness and ergodicity of these invariant measures assuming that <span><math><mi>Y</mi></math></span> is topologically recurrent, topologically irreducible, and weak Feller.</div><div>We give applications to random walks in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, which we regard as “stationary” Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between <span><math><mrow><mo>−</mo><mi>∞</mi></mrow></math></span> and <span><math><mrow><mo>+</mo><mi>∞</mi></mrow></math></span>. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings of a random walk with zero mean and finite variance of increments.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104668"},"PeriodicalIF":1.1,"publicationDate":"2025-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942254","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":"Continuum graph dynamics via population dynamics: Well-posedness, duality and equilibria","authors":"Andreas Greven ,&nbsp;Frank den Hollander ,&nbsp;Anton Klimovsky ,&nbsp;Anita Winter","doi":"10.1016/j.spa.2025.104670","DOIUrl":"10.1016/j.spa.2025.104670","url":null,"abstract":"<div><div>This paper introduces <em>graphemes</em>, a novel framework for constructing and analysing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which are well-suited for studying static dense graphs and which are closely related to the Aldous–Hoover representation of exchangeable random graphs, graphemes allow for a modelling of the full space–time evolution of <em>dynamic</em> dense graphs, beyond the exchangeability and the subgraph frequencies used in graphon theory. A grapheme is defined as an equivalence class of triples, consisting of a Polish space, a symmetric <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span>-valued connection function on that space (representing edges connecting vertices), and a sampling probability measure.</div><div>We focus on graphemes embedded in <em>ultrametric</em> spaces, where the ultrametric encodes the <em>genealogy</em> of the graph evolution, thereby drawing a direct connection to population genetics. The grapheme framework emphasises the embedding, in particular, in Polish spaces, and uses stronger notions of equivalence (homeomorphism and isometry) than the exchangeability underlying the Aldous–Hoover representation. We construct grapheme-valued Markov processes that arise as limits of finite graph evolutions, driven by rules analogous to the Fleming–Viot, Dawson–Watanabe and McKean–Vlasov processes from population genetics. We establish that these grapheme dynamics are characterised by well-posed martingale problems, leading to strong Markov processes with the Feller property and continuous paths (i.e., diffusions). We further derive duality relations by using coalescent processes, and identify the equilibria of dynamic graphemes, showing that these are linked to classical distributions arising in population genetics and can therefore be non-trivial.</div><div>Our approach extends and modifies previous work on graphon dynamics (Athreya et al., 2021), by providing a more general framework that includes a natural representation of the history of the graph. This allows for a rigorous treatment of the dynamics via martingale problems, and yields a characterisation of non-trivial equilibria.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104670"},"PeriodicalIF":1.1,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907554","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":"Walsh spider diffusions as time changed multi-parameter processes","authors":"Erhan Bayraktar ,&nbsp;Jingjie Zhang ,&nbsp;Xin Zhang","doi":"10.1016/j.spa.2025.104672","DOIUrl":"10.1016/j.spa.2025.104672","url":null,"abstract":"<div><div>Inspired by allocation strategies in multi-armed bandit model, we propose a pathwise construction of Walsh spider diffusions. For any infinitesimal generator on a star shaped graph, there exists a unique time change associated with a multi-parameter process such that the time change of this multi-parameter process is the desired diffusion. The time change has an interpretation of time allocation of the process on each edge, and it can be derived explicitly from a family of equations.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104672"},"PeriodicalIF":1.1,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895158","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":"Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space","authors":"Holger Sambale ,&nbsp;Christoph Thäle ,&nbsp;Tara Trauthwein","doi":"10.1016/j.spa.2025.104671","DOIUrl":"10.1016/j.spa.2025.104671","url":null,"abstract":"<div><div>Consider a stationary Poisson process <span><math><mi>η</mi></math></span> in the <span><math><mi>d</mi></math></span>-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set <span><math><mi>η</mi></math></span> as follows. First, each point <span><math><mrow><mi>x</mi><mo>∈</mo><mi>η</mi></mrow></math></span> is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until <span><math><mi>x</mi></math></span> is contained in the convex hull of the points already connected to <span><math><mi>x</mi></math></span>. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104671"},"PeriodicalIF":1.1,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143902472","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":"Nonparametric estimation of the transition density function for diffusion processes","authors":"Fabienne Comte ,&nbsp;Nicolas Marie","doi":"10.1016/j.spa.2025.104667","DOIUrl":"10.1016/j.spa.2025.104667","url":null,"abstract":"<div><div>We assume that we observe <span><math><mrow><mi>N</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> independent copies of a diffusion process on a time-interval <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>T</mi><mo>]</mo></mrow></math></span>. For a given time <span><math><mrow><mi>t</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow></mrow></math></span>, we estimate the transition density <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mo>.</mo><mo>)</mo></mrow></mrow></math></span>, namely the conditional density of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></msub></math></span> given <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mi>x</mi></mrow></math></span>, under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein–Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104667"},"PeriodicalIF":1.1,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882892","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}