Numerical Methods for Backward Stochastic Differential Equations in High Dimensions

Abstract

We consider the numerical solutions of the following backward stochastic dif- ferential equation(BSDE), which are of great importance in financial mathematics, stochastic control, stock markets, turbulent fluid flow ,etc. We propose a stable multi-step scheme on time-space grids for solving BSDEs. The integrands are approximated, in the time direction, by using Lagrange interpolating polynomials with values at multi- time levels. In the spatial directions, we construct a new kind of stochastic pro- cess, called Gauss-Hermite process, by which the integrands are approximated accurately and the spatial complexity can be reduced significantly. For high- dimensional problems, the sparse grids are employed to interpolate the solutions in the spatial directions, such that the high-dimensional BSDEs can be solved much more efficiently with relatively high accuracy.