A representation for the derivative with respect to the initial data of the solution of an SDE with a non-regular drift and a Gaussian noise

TL;DR

This paper develops a generalized derivative representation for solutions of multidimensional SDEs with non-regular drifts and Gaussian noise, extending classical results to less smooth cases.

Contribution

It introduces a novel representation of the derivative of SDE solutions with respect to initial data for non-smooth drifts, using continuous additive functionals.

Findings

Existence of a generalized derivative for SDE solutions with bounded variation drifts.

Representation of the derivative as a solution to a linear SDE with variable coefficients.

Extension of classical derivative formulas to non-regular drift scenarios.

Abstract

We consider a multidimensional SDE with a Gaussian noise and a drift vector being a vector function of bounded variation. We prove the existence of generalized derivative of the solution with respect to the initial conditions and represent the derivative as a solution of a linear SDE with coefficients depending on the initial process. The representation obtained is a natural generalization of the expression for the derivative in the smooth case. The theory of continuous additive functionals is used.