A numerical Fourier cosine expansion method with higher order Taylor schemes for fully coupled FBSDEs

TL;DR

This paper introduces a higher-order numerical scheme for coupled forward-backward stochastic differential equations, utilizing advanced Taylor schemes and Fourier cosine expansions to achieve improved convergence rates in both strong and weak senses.

Contribution

It presents the first fully implementable higher-order scheme for coupled FBSDEs that attains strong convergence of order 1, combining Taylor schemes with the COS method.

Findings

Achieves strong convergence order 1 for coupled FBSDEs

Demonstrates higher-order convergence in numerical experiments

Applicable to both decoupled and fully-coupled equations

Abstract

A higher-order numerical method is presented for scalar valued, coupled forward-backward stochastic differential equations. Unlike most classical references, the forward component is not only discretized by an Euler-Maruyama approximation but also by higher-order Taylor schemes. This includes the famous Milstein scheme, providing an improved strong convergence rate of order 1; and the simplified order 2.0 weak Taylor scheme exhibiting weak convergence rate of order 2. In order to have a fully-implementable scheme in case of these higher-order Taylor approximations, which involve the derivatives of the decoupling fields, we use the COS method built on Fourier cosine expansions to approximate the conditional expectations arising from the numerical approximation of the backward component. Even though higher-order numerical approximations for the backward equation are deeply studied in the literature, to the best of our understanding, the present numerical scheme is the first which achieves strong convergence of order 1 for the whole coupled system, including the forward equation, which is often the main interest in applications such as stochastic control. Numerical experiments demonstrate the proclaimed higher-order convergence, both in case of strong and weak convergence rates, for various equations ranging from decoupled to the fully-coupled settings.