On Numerical Approximations of Forward-Backward Stochastic Differential Equations

A numerical method for a class of forward-backward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the Four Step Scheme (Douglas-Ma-Protter, 1996) but with a Hermite-spectral method to approximate the solution to the decoupled quasilinear PDE on the whole space. A rigorous synthetic error analysis is carried out for a fully discretized scheme, namely a first-order scheme in time and a Hermite-spectral scheme in space, of the FBSDEs. Equally important, a systematical numerical comparison is made between several schemes for the resulting decoupled forward SDE, including a stochastic version of the Adams-Bashforth scheme. It is shown that the stochastic version of the Adams-Bashforth scheme coupled with the Hermite-spectral method leads to a convergence rate of 3 2 (in time) which is better than those in previously published work for the FBSDEs.