A STABLE MULTISTEP SCHEME FOR SOLVING BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper we propose a stable multistep scheme on time-space grids for solving backward stochastic differential equations. In our scheme, the integrands, which are conditional mathematical expectations derived from the original equations, are approximated by using Lagrange interpolating polynomials with values of the integrands at multiple time levels. They are then numerically evaluated using the Gauss-Hermite quadrature rules and polynomial interpolations on the spatial grids. Error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions. Finally, various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme.