377 research outputs found
Stochastic functional differential equations driven by L\'{e}vy processes and quasi-linear partial integro-differential equations
In this article we study a class of stochastic functional differential
equations driven by L\'{e}vy processes (in particular, -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 L\'{e}vy 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 with ), 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.Comment: 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
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