A representation for the derivative with respect to the initial data of the solution of an sde with a non-regular drift

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 obtained representation is a natural generalization of the expression for the derivative in the smooth case. In the proof we use the results on continuous additive functionals.