Backward SDEs and Cauchy problem for semilinear equations in divergence form

Abstract

We extend the definition of solutions of backward stochastic differential equations lo the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces.