On the Bounded Variation of the Flow of Stochastic Differential Equation

Abstract

We consider a stochastic differential equation, driven by a Brownian motion, with non Lipschitz coefficients. We consider the class BV which is larger than Sobolev space and got a sufficient condition for a solution of stochastic differential equation to belong to the class BV. As a consequence we prove that the corresponding flow is, almost surely, almost every where derivable with respect to initial data. The result is a partial extension of the result of N. Bouleau and F. Hirsch on the derivability, with respect to the initial data, of the solution of a stochastic differential equation with Lipschitz coefficients.