Stochastic Processes and their Applications最新文献

{"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}

{"title":"A functional central limit theorem for weighted occupancy processes of the Karlin model","authors":"Jaime Garza,&nbsp;Yizao Wang","doi":"10.1016/j.spa.2025.104665","DOIUrl":"10.1016/j.spa.2025.104665","url":null,"abstract":"<div><div>A functional central limit theorem is established for weighted occupancy processes of the Karlin model. The weighted occupancy processes take the form of, with <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow></msub></math></span> denoting the number of urns with <span><math><mi>j</mi></math></span>-balls after the first <span><math><mi>n</mi></math></span> samplings, <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></math></span> for a prescribed sequence of real numbers <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>j</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>. The main applications are limit theorems for random permutations induced by Chinese restaurant processes with <span><math><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></math></span>-seating with <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>θ</mi><mo>&gt;</mo><mo>−</mo><mi>α</mi></mrow></math></span>. An example is briefly mentioned here, and full details are provided in an accompanying paper.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104665"},"PeriodicalIF":1.1,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882890","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":"A propagation of chaos result for weakly interacting nonlinear Snell envelopes","authors":"Boualem Djehiche ,&nbsp;Roxana Dumitrescu ,&nbsp;Jia Zeng","doi":"10.1016/j.spa.2025.104669","DOIUrl":"10.1016/j.spa.2025.104669","url":null,"abstract":"<div><div>In this article, we establish a propagation of chaos result for weakly interacting nonlinear Snell envelopes which converge to a class of mean-field reflected backward stochastic differential equations (BSDEs) with jumps and right-continuous and left-limited obstacle, where the mean-field interaction in terms of the distribution of the <span><math><mi>Y</mi></math></span>-component of the solution enters both the driver and the lower obstacle. Under mild Lipschitz and integrability conditions on the coefficients, we prove existence and uniqueness of the solution to both the mean-field reflected BSDEs with jumps and the corresponding system of weakly interacting particles by using a new approach relying on the characterization of the solution of a mean-field reflected BSDE in terms of a nonlinear optimal stopping problem of mean-field type. We then provide a propagation of chaos result for the whole solution <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>,</mo><mi>U</mi><mo>,</mo><mi>K</mi><mo>)</mo></mrow></math></span>, which requires new technical results due to the dependence of the obstacle on the solution and the presence of jumps. In particular, we obtain a new law of large number type result for right-continuous and left-limited processes.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104669"},"PeriodicalIF":1.1,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882891","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":"Ladder costs for random walks in Lévy random media","authors":"Alessandra Bianchi ,&nbsp;Giampaolo Cristadoro ,&nbsp;Gaia Pozzoli","doi":"10.1016/j.spa.2025.104666","DOIUrl":"10.1016/j.spa.2025.104666","url":null,"abstract":"<div><div>We consider a random walk <span><math><mi>Y</mi></math></span> moving on a <em>Lévy random medium</em>, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> and length <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow></msub><mrow><mo>(</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>T</mi></math></span> is the first-passage time of <span><math><mi>Y</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. The study relies on the construction of a broader class of processes, denoted <em>Random Walks in Random Scenery on Bonds</em> (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of <span><math><mi>Z</mi></math></span>, corresponding to the two possible crossing directions of that bond. A random walk <span><math><mi>S</mi></math></span> on <span><math><mi>Z</mi></math></span> with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time <span><math><mi>T</mi></math></span>. This setting will be applied to obtain results for the laws of the first-ladder length and height of <span><math><mi>Y</mi></math></span>. The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk <span><math><mi>S</mi></math></span>. All these results are easily generalized to the entire sequence of ladder variables.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"188 ","pages":"Article 104666"},"PeriodicalIF":1.1,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143898835","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":"Local asymptotic properties for the growth rate of a jump-type CIR process","authors":"Mohamed Ben Alaya ,&nbsp;Ahmed Kebaier ,&nbsp;Gyula Pap ,&nbsp;Ngoc Khue Tran","doi":"10.1016/j.spa.2025.104664","DOIUrl":"10.1016/j.spa.2025.104664","url":null,"abstract":"<div><div>In this paper, we consider a one-dimensional jump-type Cox–Ingersoll–Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104664"},"PeriodicalIF":1.1,"publicationDate":"2025-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867753","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":"Speed of convergence and moderate deviations of FPP on random geometric graphs","authors":"Lucas R. de Lima ,&nbsp;Daniel Valesin","doi":"10.1016/j.spa.2025.104652","DOIUrl":"10.1016/j.spa.2025.104652","url":null,"abstract":"<div><div>This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104652"},"PeriodicalIF":1.1,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867754","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":"Lp-solutions of multi-dimensional BSDEs with mean reflection","authors":"Yue Niu ,&nbsp;Baoyou Qu ,&nbsp;Falei Wang","doi":"10.1016/j.spa.2025.104663","DOIUrl":"10.1016/j.spa.2025.104663","url":null,"abstract":"<div><div>The present paper focuses on the investigation of multi-dimensional mean reflected backward stochastic differential equations (BSDEs) in a possibly non-convex reflection domain, whose generator also depends on the marginal probability distributions of the solution <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math></span>. Our main idea is to decompose the mean reflected BSDE into a BSDE and a deterministic Skorokhod problem. Then, utilizing <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-estimates for BSDEs and Skorokhod problems, we explore the solvability of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-solutions (<span><math><mrow><mi>p</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>) through fixed-point argument and an approximation approach under both inward normal and oblique reflection scenarios.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104663"},"PeriodicalIF":1.1,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851716","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}