A convolution method for numerical solution of backward stochastic differential equations

TL;DR

This paper introduces a Fourier-based numerical method for solving backward stochastic differential equations, utilizing FFT for efficient computation and addressing error control, with applications demonstrated in finance.

Contribution

A novel Fourier analysis-based approach for numerically solving BSDEs, extending to FBSDEs and reflected variants, with error analysis and financial applications.

Findings

Effective in solving BSDEs with controlled errors

Extensible to FBSDEs and reflected FBSDEs

Demonstrated good performance in financial models

Abstract

We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed and a local error analysis is provided. We consider the extension of the method to forward-backward stochastic differential equations (FBSDEs) and reflected FBSDEs. Numerical examples are considered from finance demonstrating the performance of the method.