Zhang $L^2$-Regularity for the solutions of Backward Doubly Stochastic Differential Equations under globally Lipschitz continuous assumptions

We prove an $L^2$-regularity result for the solutions of Forward Backward Doubly Stochastic Differentiel Equations (FBDSDEs in short) under globally Lipschitz continuous assumptions on the coefficients. Therefore, we extend the well known regularity results established by Zhang (2004) for Forward Backward Stochastic Differential Equations (FBSDEs in short) to the doubly stochastic framework. To this end, we prove (by Malliavin calculus) a representation result for the martingale component of the solution of the F-BDSDE under the assumption that the coefficients are continuous in time and continuously differentiable in space with bounded partial derivatives. As an (important) application of our $L^2$-regularity result, we derive the rate of convergence in time for the (Euler time discretization based) numerical scheme for FBDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.