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 3/2 ( in time) which is better than those in previously published work for the FBSDEs.