On differentiability of stochastic flow for a multidimensional SDE with discontinuous drift

TL;DR

This paper investigates the differentiability of solutions to a multidimensional stochastic differential equation with discontinuous drift, providing a representation for how solutions depend on initial conditions.

Contribution

It introduces a novel representation for the derivative of the solution with respect to initial data in SDEs with discontinuous drift.

Findings

Derived a representation for the derivative of the solution.

Extended differentiability analysis to SDEs with discontinuous drift.

Applicable to multidimensional SDEs with bounded variation drifts.

Abstract

We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.