  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.
