2011/06/30 by Xicheng Zhang
Computer Science · Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Bounded function #Differential equation #Lipschitz continuity #Mathematical analysis #Mathematics #Method of characteristics #Nonlinear system #Partial differential equation #Physics #Stability and Controllability of Differential Equations #Stochastic differential equation #Stochastic partial differential equation #Stochastic processes and financial applications #math.AP #math.PR
paper · pdf · doi:10.1214/12-aap851
published as Annals of Applied Probability 2012, Vol. 22, No. 6, 2505-2538 · Published in at http://dx.doi.org/10.1214/12-AAP851 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2012/11/23 · arxiv created 2012/11/29 · arxiv updated 2012/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, α-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 Lévy 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 Δα/2 with α∈(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.