Numerical scheme for backward doubly stochastic differential equations

We study a discrete-time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations (FBDSDEs). Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the step of time discretization, $\mathrm{\mid }\pi \mathrm{\mid }$ goes to zero. The rate of convergence is exactly equal to $\mathrm{\mid }\pi {\mathrm{\mid }}^{1\mathrm{/}2}$. The proof is based on a generalization of a remarkable result on the ${}^2$-regularity of the solution of the backward equation derived by J. Zhang