Stochastic Processes and their Applications最新文献

{"title":"A class of processes defined in the white noise space through generalized fractional operators","authors":"Luisa Beghin ,&nbsp;Lorenzo Cristofaro ,&nbsp;Yuliya Mishura","doi":"10.1016/j.spa.2024.104494","DOIUrl":"10.1016/j.spa.2024.104494","url":null,"abstract":"<div><div>The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104494"},"PeriodicalIF":1.1,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319393","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":"A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens–Pitman model","authors":"Bernard Bercu ,&nbsp;Stefano Favaro","doi":"10.1016/j.spa.2024.104493","DOIUrl":"10.1016/j.spa.2024.104493","url":null,"abstract":"<div><p>The Ewens–Pitman model refers to a distribution for random partitions of <span><math><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></mrow></math></span>, which is indexed by a pair of parameters <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>θ</mi><mo>&gt;</mo><mo>−</mo><mi>α</mi></mrow></math></span>, with <span><math><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math></span> corresponding to the Ewens model in population genetics. The large <span><math><mi>n</mi></math></span> asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of partition sets and the number <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> of partition subsets of size <span><math><mi>r</mi></math></span>, for <span><math><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></math></span>. While for <span><math><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math></span> asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> only almost-sure convergences are available, with the proof for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large <span><math><mi>n</mi></math></span> asymptotic behaviours of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, providing alternative, and rigorous, proofs of the almost-sure convergences of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104493"},"PeriodicalIF":1.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270521","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":"Markov chains generating random permutations and set partitions","authors":"Dudley Stark","doi":"10.1016/j.spa.2024.104483","DOIUrl":"10.1016/j.spa.2024.104483","url":null,"abstract":"<div><p>The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on <span><math><mi>n</mi></math></span> elements proportionally to absorption probabilities <span><math><msup><mrow><mi>θ</mi></mrow><mrow><mrow><mo>|</mo><mi>π</mi><mo>|</mo></mrow></mrow></msup></math></span>, <span><math><mrow><mi>θ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, where <span><math><mrow><mo>|</mo><mi>π</mi><mo>|</mo></mrow></math></span> is the number of cycles of permutation <span><math><mi>π</mi></math></span>. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104483"},"PeriodicalIF":1.1,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304414924001893/pdfft?md5=12029ff1851856f47073a4ee02bb7a29&pid=1-s2.0-S0304414924001893-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173192","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":"Strong limit theorems for step-reinforced random walks","authors":"Zhishui Hu,&nbsp;Yiting Zhang","doi":"10.1016/j.spa.2024.104484","DOIUrl":"10.1016/j.spa.2024.104484","url":null,"abstract":"<div><p>A step-reinforced random walk is a discrete-time process with long range memory. At each step, with a fixed probability <span><math><mi>p</mi></math></span>, the positively step-reinforced random walk repeats one of its preceding steps chosen uniformly at random, and with complementary probability <span><math><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></math></span>, it has an independent increment. The negatively step-reinforced random walk follows the same reinforcement algorithm but when a step is repeated its sign is also changed. Strong laws of large numbers and strong invariance principles are established for positively and negatively step-reinforced random walks in this work. Our approach relies on two general theorems on the invariance principles for martingale difference sequences and a truncation argument. As by-products of our main results, the law of iterated logarithm and the functional central limit theorem are also obtained for step-reinforced random walks.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104484"},"PeriodicalIF":1.1,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142149346","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":"Fractional stable random fields on the Sierpiński gasket","authors":"Fabrice Baudoin ,&nbsp;Céline Lacaux","doi":"10.1016/j.spa.2024.104481","DOIUrl":"10.1016/j.spa.2024.104481","url":null,"abstract":"<div><div>We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as <span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>α</mi></mrow></msub></mrow></math></span>, where <span><math><mi>Δ</mi></math></span> is the Laplace operator on the gasket and <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> is a stable random measure. Both Neumann and Dirichlet boundary conditions for <span><math><mi>Δ</mi></math></span> are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104481"},"PeriodicalIF":1.1,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357534","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":"A stochastic analysis of particle systems with pairing","authors":"Vincent Fromion ,&nbsp;Philippe Robert ,&nbsp;Jana Zaherddine","doi":"10.1016/j.spa.2024.104480","DOIUrl":"10.1016/j.spa.2024.104480","url":null,"abstract":"<div><p>Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each particle follows a random path in the medium, when a particle and an agent meet, they may bind and form a pair which has some specific functional properties. Such a pair is also subject to random events and it splits after some random amount of time. In a stochastic context, using a Markovian model for the vector of the number of paired particles, and by taking the total number of particles as a scaling parameter, we study the asymptotic behavior of the time evolution of the number of paired particles. Two scenarios are investigated: one with a large but fixed number of agents, and the other one, the dynamic case, when agents are created at a bounded rate and may die after some time when they are not paired. A first order limit theorem is established for the time evolution of the system in both cases. The proof of an averaging principle of the dynamic case is one of the main contributions of the paper. The impact of dynamical arrivals of agents on the level of pairing of the system is discussed.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104480"},"PeriodicalIF":1.1,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136916","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":"Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations","authors":"Yushi Hamaguchi","doi":"10.1016/j.spa.2024.104482","DOIUrl":"10.1016/j.spa.2024.104482","url":null,"abstract":"<div><p>We introduce a new framework of Markovian lifts of stochastic Volterra integral equations (SVIEs for short) with completely monotone kernels. We define the state space of the Markovian lift as a separable Hilbert space which incorporates the singularity or regularity of the kernel into the definition. We show that the solution of an SVIE is represented by the solution of a lifted stochastic evolution equation (SEE for short) defined on the Hilbert space and prove the existence, uniqueness and Markov property of the solution of the lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and some consequent properties for the Markov semigroup associated with the Markovian lift via the asymptotic coupling method.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104482"},"PeriodicalIF":1.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142149345","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":"Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients","authors":"Chenxu Pang,&nbsp;Xiaojie Wang","doi":"10.1016/j.spa.2024.104467","DOIUrl":"10.1016/j.spa.2024.104467","url":null,"abstract":"<div><p>In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for the required target accuracy <span><math><mi>ϵ</mi></math></span>. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104467"},"PeriodicalIF":1.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164121","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":"Large deviations for slow–fast processes on connected complete Riemannian manifolds","authors":"Yanyan Hu ,&nbsp;Richard C. Kraaij ,&nbsp;Fubao Xi","doi":"10.1016/j.spa.2024.104478","DOIUrl":"10.1016/j.spa.2024.104478","url":null,"abstract":"<div><p>We consider a class of slow–fast processes on a connected complete Riemannian manifold <span><math><mi>M</mi></math></span>. The limiting dynamics as the scale separation goes to <span><math><mi>∞</mi></math></span> is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with Hamilton–Jacobi–Bellman (HJB) equation techniques. Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on <span><math><mi>M</mi></math></span> and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104478"},"PeriodicalIF":1.1,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304414924001844/pdfft?md5=48570b477d6d4ad61f1d0e5520f39079&pid=1-s2.0-S0304414924001844-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168200","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":"Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions and past light-cone property","authors":"Juan J. Jiménez","doi":"10.1016/j.spa.2024.104479","DOIUrl":"10.1016/j.spa.2024.104479","url":null,"abstract":"<div><p>In this article, we study the stochastic wave equation in spatial dimensions <span><math><mrow><mi>d</mi><mo>≤</mo><mn>2</mn></mrow></math></span> with multiplicative Lévy noise that can have infinite <span><math><mi>p</mi></math></span>th moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the <span><math><mi>p</mi></math></span>-integrability of the Lévy measure <span><math><mi>ν</mi></math></span> for the region corresponding to the small jumps of the noise. For <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span>, there are no restrictions on <span><math><mi>ν</mi></math></span>. For <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span>, we assume that there exists a value <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> for which <span><math><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mrow><mo>{</mo><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>≤</mo><mn>1</mn><mo>}</mo></mrow></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi>ν</mi><mrow><mo>(</mo><mi>d</mi><mi>z</mi><mo>)</mo></mrow><mo>&lt;</mo><mo>+</mo><mi>∞</mi></mrow></math></span>.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"178 ","pages":"Article 104479"},"PeriodicalIF":1.1,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142149344","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}