Abstract

We present a Fourier analysis approach to numerical solution of forward-backward stochastic differential equations (FBSDEs) and propose two implementations. Using the Euler time discretization for backward stochastic differential equations (BSDEs), Fourier analysis allows to express the conditional expectations included in the time discretization in terms of Fourier integrals. The space discretization of these integrals then leads to expressions involving discrete Fourier transforms (DFTs) so that the FFT algorithm can be used. We quickly presents the convolution method on a uniform space grid. Locally, this first implementation produces a truncation error, a space discretization error and an additional extrapolation error. Even if the extrapolation error is convergent in time, the resulting absolute error may be high at the boundaries of the uniform space grid. In order to solve this problem, we propose a tree-like grid for the space discretization which suppresses the extrapolation error leading to a globally convergent numerical solution for the BSDE. The method is then extended to FBSDEs with bounded coefficients, reflected FBSDEs and higher order time discretizations of FBSDEs. Numerical examples from finance illustrate its performance.