Empirical Regression Method for Backward Doubly Stochastic Differential Equations

In this paper we design a numerical scheme for approximating backward doubly stochastic differential equations which represent a solution to stochastic partial differential equations. We first use a time discretization and then we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion B. The coefficients are evaluated through an empirical regression scheme, which is performed conditionally to B. We establish nonasymptotic error estimates, conditionally to B, and deduce how to tune parameters to obtain a convergence conditionally and unconditionally to B. We provide numerical experiments as well.