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

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.