Fixed point technique for a class of backward stochastic differential equations

We establish a new theorem on the existence and uniqueness of the adapted solution to backward stochastic differential equations under some weaker conditions than the Lipschitz one. The extension is based on Athanassov s condition for ordinary differential equations. In order to prove the existence of the solutions we use a fixed point technique based on Schauder s fixed point theorem. Also, we study some regularity properties of the solution for this class of stochastic differential equations.