Empirical regression method for backward doubly stochastic differential equations

In this paper we design a numerical scheme for approximating backward doubly stochastic differen- tial 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 regres- sion 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.