Mean square rate of convergence for random walk approximation of forward-backward SDEs

Abstract: Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,$\alpha$. The proof relies on an approximative representation of Z n and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by stochastic methods.