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We prove an $L^2$-regularity result for the solutions of Forward BackwardDoubly Stochastic Differentiel Equations (FBDSDEs in short) under globallyLipschitz continuous assumptions on the coefficients. Therefore, we extend thewell known regularity results established by Zhang (2004) for Forward BackwardStochastic Differential Equations (FBSDEs in short) to the doubly stochasticframework. To this end, we prove (by Malliavin calculus) a representationresult for the martingale component of the solution of the F-BDSDE under theassumption that the coefficients are continuous in time and continuouslydifferentiable in space with bounded partial derivatives. As an (important)application of our $L^2$-regularity result, we derive the rate of convergencein time for the (Euler time discretization based) numerical scheme for FBDSDEsproposed by Bachouch et al. (2016) under only globally Lipschitz continuousassumptions.

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