A Fourier Transform Method for Solving Backward Stochastic Differential Equations

We propose a method based on the Fourier transform for numerically solving backward stochastic differential equations. Time discretization is applied to the forward equation of the state variable as well as the backward equation to yield a recursive system with terminal conditions. By assuming the integrability of the functions in the terminal conditions and applying truncation, the solutions of the system are shown to be integrable and we derive recursions in the Fourier space. The fractional FFT algorithm is applied to compute the Fourier and inverse Fourier transforms. We showcase the efficiency of our method through various numerical examples.