STOCHASTIC VOLTERRA EQUATIONS FOR THE LOCAL TIMES OFSPECTRALLY POSITIVE STABLE PROCESSES

• 成果类型: Article
• 署名作者: Xu, Wei
• 署名单位: Beijing Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2017
• 发表日期: 2024
• 页码: 2733-2798
• 关键词: branching-processes; LIMIT-THEOREMS; SCALING LIMITS; Levy processes; rough; continuity; driven; FLOWS

摘要

This paper is concerned with the evolution dynamics of local times ofa spectrally positive stable process in the spatial direction. The main resultsstate that conditioned on the finiteness of the first time at which the local timeat zero exceeds a given value, the local times at positive half line are equal indistribution to the unique solution of a stochastic Volterra equation driven bya Poisson random measure whose intensity coincides with the L & eacute;vy measure.This helps us to provide not only a simple proof for the H & ouml;lder regularity,but also a uniform upper bound for all moments of the H & ouml;lder coefficientas well as a maximal inequality for the local times. Moreover, based on thisstochastic Volterra equation, we extend the method of duality to establish anexponential-affine representation of the Laplace functional in terms of theunique solution of a nonlinear Volterra integral equation associated with theLaplace exponent of the stable process.