Stochastic Processes and their Applications最新文献

{"title":"Regularization effects of time integration on Gaussian process functionals","authors":"Takafumi Amaba ,&nbsp;Marie Kratz","doi":"10.1016/j.spa.2025.104761","DOIUrl":"10.1016/j.spa.2025.104761","url":null,"abstract":"<div><div>In this paper, we investigate the regularization effects, in the sense of Malliavin calculus, on functionals of Gaussian processes induced by time integration, focusing on their covariance functions. We study several examples of important covariance functions classes to verify whether they satisfy the sufficient conditions proposed for regularization. Additionally, we derive a weak implication for the smoothness of level-crossing functionals.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104761"},"PeriodicalIF":1.2,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144889613","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":"On favourite sites of a random walk in moderately sparse random environment","authors":"Alicja Kołodziejska","doi":"10.1016/j.spa.2025.104760","DOIUrl":"10.1016/j.spa.2025.104760","url":null,"abstract":"<div><div>We study the favourite sites of a random walk evolving in a sparse random environment on the set of integers. The walker moves symmetrically apart from some randomly chosen sites where we impose random drift. We prove annealed limit theorems for the time the walk spends in its favourite sites in two cases. The first one, in which it is the distribution of the drift that determines the limiting behaviour of the walk, is a generalization of known results for a random walk in i.i.d. random environment. In the second case a new behaviour appears, caused by the sparsity of the environment.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104760"},"PeriodicalIF":1.2,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766636","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":"Stochastic approximation with two time scales: The general case","authors":"Vivek S. Borkar","doi":"10.1016/j.spa.2025.104759","DOIUrl":"10.1016/j.spa.2025.104759","url":null,"abstract":"<div><div>Two time scale stochastic approximation is analyzed when the iterates on either or both time scales do not necessarily converge.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104759"},"PeriodicalIF":1.2,"publicationDate":"2025-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739335","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":"Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs","authors":"Xinying Li,&nbsp;Yaqi Zhang,&nbsp;Shengjun Fan","doi":"10.1016/j.spa.2025.104758","DOIUrl":"10.1016/j.spa.2025.104758","url":null,"abstract":"<div><div>This study focuses on a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time <span><math><mi>τ</mi></math></span> taking values in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mi>∞</mi><mo>]</mo></mrow></math></span>. The generator <span><math><mi>g</mi></math></span> satisfies a stochastic monotonicity condition in the first unknown variable <span><math><mi>y</mi></math></span> and a stochastic Lipschitz continuity condition in the second unknown variable <span><math><mi>z</mi></math></span>, and it can have a more general growth with respect to <span><math><mi>y</mi></math></span> than the classical one stated in (H5) of Briand et al. (2003). Without imposing any restriction of finite moment on the stochastic coefficients, we establish a general existence and uniqueness result for the weighted solution of such BSDE in a proper weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-space with a suitable weighted factor. This result is proved via some innovative ideas and delicate analytical techniques, and it unifies and strengthens some existing works on BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz generators, and BSDEs with deterministic Lipschitz/monotonicity generators. Then, a continuous dependence property and a stability theorem for the weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-solutions are given. We also derive the nonlinear Feynman–Kac formulas for both parabolic and elliptic PDEs in our context.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104758"},"PeriodicalIF":1.2,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724784","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":"Approximation of birth–death processes","authors":"Liping Li","doi":"10.1016/j.spa.2025.104756","DOIUrl":"10.1016/j.spa.2025.104756","url":null,"abstract":"<div><div>A birth–death process is a special type of continuous-time Markov chains with minimal state space <span><math><mi>N</mi></math></span>. Its resolvent matrix can be fully characterized by a set of parameters <span><math><mrow><mo>(</mo><mi>γ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>ν</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>γ</mi></math></span> and <span><math><mi>β</mi></math></span> are non-negative constants, and <span><math><mi>ν</mi></math></span> is a positive measure on <span><math><mi>N</mi></math></span>. By employing the Ray-Knight compactification, the birth–death process can be realized as a càdlàg process with strong Markov property on the one-point compactification space <span><math><msub><mrow><mover><mrow><mi>N</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>∂</mi></mrow></msub></math></span>, which includes an additional cemetery point <span><math><mi>∂</mi></math></span>. In a certain sense, the three parameters that determine the birth–death process correspond to its killing, reflecting, and jumping behaviors at <span><math><mi>∞</mi></math></span> used for the one-point compactification, respectively.</div><div>In general, providing a clear description of the trajectories of a birth–death process, especially in the pathological case where <span><math><mrow><mrow><mo>|</mo><mi>ν</mi><mo>|</mo></mrow><mo>=</mo><mi>∞</mi></mrow></math></span>, is challenging. This paper aims to address this issue by studying the birth–death process using approximation methods. Specifically, we will approximate the birth–death process with simpler birth–death processes that are easier to comprehend. For two typical approximation methods, our main results establish the weak convergence of a sequence of probability measures, which are induced by the approximating processes, on the space of all càdlàg functions. This type of convergence is significantly stronger than the convergence of transition matrices typically considered in the theory of continuous-time Markov chains.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104756"},"PeriodicalIF":1.1,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144714535","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":"Rough path lifts of Banach space-valued Gaussian processes","authors":"A.A. Kalinichenko","doi":"10.1016/j.spa.2025.104739","DOIUrl":"10.1016/j.spa.2025.104739","url":null,"abstract":"<div><div>Under certain assumptions on a Gaussian process taking values in a separable Banach space, we construct its lift to a geometric rough path. The lift is natural in the sense that for any sequence of piece-wise linear approximations to the original process, their signatures converge to the lifted path in a suitable metric. This extends to infinite dimensions the known results in Euclidean spaces. Examples of processes satisfying our conditions include the infinite-dimensional analogues of Brownian motion, fractional Brownian motion with Hurst parameter <span><math><mrow><mi>H</mi><mo>∈</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></mrow></mrow></math></span>, Ornstein–Uhlenbeck process. As a by-product of our methods, we also provide a construction for Ito–Skorokhod integrals of these processes, which might be of independent interest.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104739"},"PeriodicalIF":1.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680189","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":"Parisi PDE and convexity for vector spins","authors":"Hong-Bin Chen","doi":"10.1016/j.spa.2025.104746","DOIUrl":"10.1016/j.spa.2025.104746","url":null,"abstract":"<div><div>We consider mean-field vector spin glasses with self-overlap correction. The limit of free energy is known to be the Parisi formula, which is an infimum over matrix-valued paths. We decompose such a path into a Lipschitz matrix-valued path and the quantile function of a one-dimensional probability measure. For such a pair, we associate a Parisi PDE generalized for vector spins. Under mild conditions, we rewrite the Parisi formula in terms of solutions of the PDE. Moreover, for each fixed Lipschitz path, the Parisi functional is strictly convex over probability measures.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104746"},"PeriodicalIF":1.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680238","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":"Scaling limits for interactive Hawkes shot noise processes","authors":"Bo Li ,&nbsp;Guodong Pang","doi":"10.1016/j.spa.2025.104748","DOIUrl":"10.1016/j.spa.2025.104748","url":null,"abstract":"<div><div>We introduce an interactive Hawkes shot noise process, in which the shot noise process has a Hawkes arrival process whose intensity depends on the state of the shot noise process via the self-exciting function. Namely, the shot noise process and the Hawkes process are interactive. We prove a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the joint dynamics of shot noise process and the Hawkes process, and characterize the effect of the interaction between them. The FLLN limit is determined by a nonlinear function determined through an integral equation. The diffusion limit is a two-dimensional interactive stochastic differential equation driven by two independent time-changed Brownian motions. The limit of the CLT-scaled shot noise process itself can be also expressed equivalently in distribution as an Ornstein–Uhlenbeck process with time-dependent parameters, unlike being a Brownian motion in the standard case without interaction. The limit of the CLT-scaled Hawkes counting process can be expressed as a sum of two independent terms, one as a time-changed Brownian motion (just as the standard case), and the other as a (Volterra type) Gaussian process represented by an Itô integral with another time-changed Brownian motion, capturing the effect of the interaction in the self-exciting function with the state of the shot noise process. To prove the joint convergence of the co-dependent Hawkes and shot noise processes, the standard techniques for Hawkes processes using the immigration-birth representations and the associated renewal equations are no longer applicable. We develop novel techniques by constructing representations for the LLN and CLT scaled processes that resemble the limits together with the associated residual terms, and then use a localization technique together with some martingale properties to prove the residual terms converge to zero and hence the joint convergence of the scaled processes. We also consider an extension of our model, an interactive marked Hawkes shot noise process, where the intensity of the Hawkes arrivals also depends on an exogenous noise, and present the corresponding FLLN and FCLT limits.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104748"},"PeriodicalIF":1.1,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696962","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":"Multidimensional sticky Brownian motions: Heavy traffic limit and rough tail asymptotics","authors":"Hongshuai Dai ,&nbsp;Yiqiang Q. Zhao","doi":"10.1016/j.spa.2025.104743","DOIUrl":"10.1016/j.spa.2025.104743","url":null,"abstract":"<div><div>Inspired by the concept of sticky Brownian motion on the half-line, we investigate a time-changed semimartingale reflecting Brownian motion in the orthant, which we refer to as multidimensional sticky Brownian motion. We first show that it can be obtained as a natural diffusion approximation for a certain tandem queue with exceptional arrival rates. Furthermore, we examine the tail dependence structure of the joint stationary distribution. Under some mild conditions, we derive rough tail asymptotics for the joint stationary distribution. Finally, in some special cases, we present the exact tail asymptotics of the joint stationary distribution.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104743"},"PeriodicalIF":1.1,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696963","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":"On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary","authors":"V. Liang,&nbsp;K. Borovkov","doi":"10.1016/j.spa.2025.104742","DOIUrl":"10.1016/j.spa.2025.104742","url":null,"abstract":"<div><div>The paper analyses the sensitivity of the finite time horizon boundary non-crossing probability <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></math></span> of a general time-inhomogeneous, one-dimensional diffusion process to perturbations of the boundary <span><math><mi>g</mi></math></span>. We prove that, for time-dependent boundaries <span><math><mrow><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span> this probability is Gâteaux differentiable in directions <span><math><mrow><mi>h</mi><mo>∈</mo><mi>H</mi><mo>∪</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and Fréchet-differentiable in directions <span><math><mrow><mi>h</mi><mo>∈</mo><mi>H</mi><mo>,</mo></mrow></math></span> where <span><math><mi>H</mi></math></span> is the Cameron–Martin space, and derive a compact representation for the derivative of <span><math><mi>F</mi></math></span>. Our results allow one to approximate <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></math></span> using boundaries <span><math><mover><mrow><mi>g</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> that are close to <span><math><mi>g</mi></math></span> and for which the computation of <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mover><mrow><mi>g</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span> is feasible. We also obtain auxiliary results of independent interest in both probability theory and PDE theory.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104742"},"PeriodicalIF":1.1,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144665965","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}