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

TL;DR

This paper investigates a class of jump-only stochastic differential equations, establishing their solutions' properties, regularity, and connections to non-local PDEs, including Harnack inequalities and recurrence behaviors.

Contribution

It introduces new stochastic characterizations, proves Harnack inequalities, and links recurrence properties with non-local PDEs for jump processes.

Findings

Proved Harnack inequality for jump operators

Established regularity of solutions to Dirichlet problems

Connected recurrence properties with non-local PDEs

Abstract

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.