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

Balint Negyesi, Cornelis W. Oosterlee

TL;DR

This work introduces a fully implementable BCOS/COS-based numerical scheme for fully-coupled forward-backward SDEs, combining higher-order Itô–Taylor discretizations for the forward SDE (including Milstein and simplified order 2.0 weak Taylor) with backward generalized theta schemes for the BSDE. By leveraging decoupling fields and closed-form characteristic functions, the method computes conditional expectations via Fourier cosine expansions and Distinct Cosine Transforms with Picard iterations handling implicit steps. The authors prove that the approach attains strong convergence order 1 for the entire coupled system and weak convergence order 2, demonstrated through numerical experiments spanning decoupled to fully coupled problems, including stochastic control. The results show significant improvements in forward and backward accuracy over Euler-based baselines, with modest additional computational cost, and robustness to Fourier truncation across problem classes.

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.

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

TL;DR

This work introduces a fully implementable BCOS/COS-based numerical scheme for fully-coupled forward-backward SDEs, combining higher-order Itô–Taylor discretizations for the forward SDE (including Milstein and simplified order 2.0 weak Taylor) with backward generalized theta schemes for the BSDE. By leveraging decoupling fields and closed-form characteristic functions, the method computes conditional expectations via Fourier cosine expansions and Distinct Cosine Transforms with Picard iterations handling implicit steps. The authors prove that the approach attains strong convergence order 1 for the entire coupled system and weak convergence order 2, demonstrated through numerical experiments spanning decoupled to fully coupled problems, including stochastic control. The results show significant improvements in forward and backward accuracy over Euler-based baselines, with modest additional computational cost, and robustness to Fourier truncation across problem classes.

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.
Paper Structure (20 sections, 3 theorems, 55 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 20 sections, 3 theorems, 55 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discrete time approximations of forward-backward stochastic differential equations
  3. Forward discretization
  4. Backward discretizations
  5. Single versus multi-step backward schemes.
  6. COS approximations
  7. Derivative approximations and the BCOS method with higher-order Taylor schemes
  8. Coefficient recovery.
  9. Picard iterations.
  10. Errors and computational complexity
  11. Error analysis.
  12. Computational complexity.
  13. Numerical experiments
  14. Example 1: decoupled FBSDE
  15. Example 2: partial coupling
  16. ...and 5 more sections

Key Result

Lemma 3.1

Consider eq:sde:discretization:decoupled, for any decoupling pair $\varphi, \zeta$, the characteristic function of $X_{n+1}^{\pi, \varphi, \zeta}$ given $X_n^{\pi, \varphi, \zeta}=x$ reads as follows

Figures (5)

  • Figure 1: Example 1 in \ref{['eq:example1']}. ($\theta_4=0, \theta_1=\theta_2=\theta_3=1/2$, $K=2^9$.)
  • Figure 2: Example 2 in \ref{['eq:example2']} with $\kappa_z=0$. Strong convergence. ($\theta_1=\theta_2=\theta_3=1/2$, $K=2^{10}$.)
  • Figure 3: Example 2 in \ref{['eq:example2']} with $\kappa_z=0$. Weak convergence at $t_0$. ($\theta_1=\theta_2=\theta_3=1/2$, $K=2^{10}$.)
  • Figure 4: Example 2 in \ref{['eq:example2']}, with $\kappa_z=10^{-2}$. ($\theta_1=\theta_2=\theta_3=1/2$, $\theta_4=-1/2$, $K=2^{10}$.)
  • Figure 5: Example 3 in \ref{['eq:example3']}. ($\theta_1=\theta_2=\theta_3=1/2$, $\theta_4=-1/2$, $K=2^{10}$.)

Theorems & Definitions (8)

  • Remark 2.1
  • Lemma 3.1: Characteristic function of Markov transitions, ruijter_numerical_2016
  • proof
  • Lemma 3.2: Integration by parts formulae with discretization \ref{['eq:sde:discretization:decoupled']}
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1