Singular SDEs with critical non-local and non-symmetric L\'evy type generator

TL;DR

This paper develops a method to construct solutions for stochastic differential equations driven by non-local, non-symmetric Lévy processes with irregular coefficients, expanding understanding of such complex stochastic systems.

Contribution

It introduces a Levi's parametrix approach to construct fundamental solutions for critical non-local operators and proves existence and uniqueness of solutions for related SDEs.

Findings

Constructed fundamental solutions for critical non-local operators.

Proved existence and uniqueness of strong solutions for the SDEs.

Extended analysis to irregular coefficient cases.

Abstract

In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the existence and uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric L\'evy type operator.