Abstract. Efficient numerical algorithms are proposed for a class of forward-backward stochastic differential equations (FBSDEs) connected with semilinear parabolic partial differential equations. As in [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940–968], the algorithms are based on the known four-step scheme for solving FBSDEs. The corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach. The derivatives of the solution u of the semilinear equation are found by finite differences. The forward equation is simulated by mean-square methods of order 1/2 and 1. Corresponding convergence theorems are proved. Along with the algorithms for FBSDEs on a fixed finite time interval, we also construct algorithms for FBSDEs with random terminal time. The results obtained are supported by numerical experiments. Key words. forward-backward stochastic differential equations, numerical integration, meansquare convergence, semilinear partial differential equations of parabolic type

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

Abstract. The method of characteristics (the averaging over the characteristicformula)andtheweak-sensenumerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. A variance reduction technique for the Monte Carlo procedures is considered. Layer methods for linear and semilinear SPDEs are constructed and the corresponding convergence theorems are proved. The approach developed is supported by numerical experiments. 1.

Abstract. The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter. 1.