Discretization of Pardoux-Peng's Backward Stochastic Differential Equations

摘要

In this paper, we present a probabilistic numerical method for solving backward stochastic differential equations introduced by E. Pardoux and S. Peng., associated with a forward S.D.E. We restrict ourselves to the case where the generator of the backward equation depends on the forward and backward processes but not on the diffusion term. First we build a Euler scheme with n steps to discretize the backward S.D.E. in time. Then we show that it is necessary to make a second discretization in space, which leads to evaluate the solution of the backward S.D.E. along N simulated trajectories of the forward process. Thus we build up an algorithm which permits to estimate the value at time zero of the solution of the backward S.D.E. with an error of order 1 + n/N in dimension 1.