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In this article we study a class of stochastic functional differentialequations driven by L\'{e}vy processes (in particular, $\alpha$-stableprocesses), and obtain the existence and uniqueness of Markov solutions insmall time intervals. This corresponds to the local solvability to a class ofquasi-linear partial integro-differential equations. Moreover, in the constantdiffusion 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 ofsemi-linear partial integro-differential equation systems under boundedLipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case(corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), based upon somegradient estimates, the existence of global solutions is established too. Inparticular, this provides a probabilistic treatment for the nonlinear partialintegro-differential equations, such as the multi-dimensional fractal Burgersequations and the fractal scalar conservation law equations.

Stochastic functional differential equations driven by L\'{e}vy processes and quasi-linear partial integro-differential equations