A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations

TL;DR

This paper introduces a Fourier-based numerical method for solving FBSDEs that uses a tree-like spatial discretization to improve convergence, stability, and enable higher-order discretizations, with demonstrated effectiveness on financial models.

Contribution

It presents a novel tree-like spatial discretization for FBSDEs that eliminates the need for spatial interpolation and achieves global convergence with explicit rates.

Findings

Achieves global convergence with explicit rates.

Extends to higher-order time discretizations.

Demonstrates effectiveness on commodity price models.

Abstract

The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs), introduced in [21], uses a uniform space grid. In this paper we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In addition to suppressing extrapolation error, leading to a globally convergent numerical solution for the FBSDE, we provide explicit convergence rates. On this alternative grid the conditional expectations involved in the time discretization of the BSDE are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm. The method is then extended to higher-order time discretizations of FBSDEs. Numerical results demonstrating convergence are presented using a commodity price model, incorporating seasonality, and forward prices.