On a Class of Stochastic Differential Equations With Jumps and Its Properties

Ari Arapostathis, Anup Biswas, Luis Caffarelli

Published 2014-01-23, updated 2014-11-06Version 2

We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator. One of the key results is the regularity of solutions of the Dirichlet problem for a class of operators with locally weakly H\"older continuous kernels.