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 [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940-968] but with a Hermite-spectral method to approximate the solution to the decoupling quasi-linear 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 $\frac 32$ (in time) which is better than those in previously published work for the FBSDEs.