On differentiability with respect to the initial data of a solution of an SDE with L\'evy noise and discontinuous coefficients

TL;DR

This paper constructs a stochastic flow for an SDE driven by Lévy noise with discontinuous drift, proving its Sobolev differentiability with respect to initial data and providing a derivative representation.

Contribution

It introduces a novel approach to analyze SDEs with Lévy noise and discontinuous coefficients, establishing Sobolev differentiability of the flow.

Findings

Flow is non-coalescing

Flow is Sobolev differentiable with respect to initial data

Derivative representation is provided

Abstract

We construct a stochastic flow generated by an SDE with L\'evy noise and a drift coefficient being a function of bounded variation on R. It is proved that this flow is non-coalescing and Sobolev differentiable with respect to initial data. The representation for the derivative is given.