STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS DRIVEN BY LEVY PROCESSES AND QUASI-LINEAR PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS
成果类型：
Article
署名作者：
Zhang, Xicheng
署名单位：
Wuhan University
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/12-AAP851
发表日期：
2012
页码：
2505-2538
关键词：
摘要：
In this article we study a class of stochastic functional differential equations driven by Levy processes (in particular, alpha-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Levy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding, to Delta(alpha/2) with alpha is an element of (1, 2]), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.