Irregular Stochastic differential equations driven by a family of Markov processes

TL;DR

This paper establishes pathwise uniqueness for strong solutions of irregular stochastic differential equations driven by non-local, non-symmetric Markov processes, utilizing heat kernel estimates and novel regularity results.

Contribution

It introduces new regularity results for generators of non-local, non-symmetric Markov processes and proves pathwise uniqueness under irregular conditions.

Findings

Proved pathwise uniqueness for a class of irregular SDEs

Developed new regularity estimates for non-local generators

Applied heat kernel estimates to stochastic equations

Abstract

Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric L\'evy type operator. Due to the extra term $1_{[0,\sigma(X_{s-},z)]}(r)$ in multiplicative noise, we need to derive some new regularity results for the generator and use a trick of mixing $L_1$ and $L_2$-estimates by Kurtz and Protter \cite{Ku-Po}.