Explicit $θ$-Schemes for Solving Anticipated Backward Stochastic Differential Equations

TL;DR

This work tackles solving anticipated BSDEs where the generator depends on future values by introducing stable explicit $\theta$-schemes that convert delay terms into current measurable quantities through conditional expectations. The authors derive reference equations on a uniform time grid, formulate backward schemes parameterized by $\theta_1,\theta_2,\theta_3$, and incorporate delay interpolation to handle $Y_{t+\delta(t)}$ and $Z_{t+\zeta(t)}$. They prove stability via a discrete Gronwall framework and establish rigorous error estimates linked to local truncation errors, with higher-order convergence attained under appropriate smoothness and $\theta$-choices (notably $\theta_i=\tfrac{1}{2}$). Numerical experiments with linear and nonlinear examples, using Gauss–Hermite quadrature and spline interpolation, confirm both the stability and the predicted convergence rates, demonstrating the schemes' efficiency and accuracy for anticipated BSDEs. The approach offers a practical path to high-order, explicit methods in a challenging stochastic delay setting, with potential impact on numerical stochastic control and financial engineering applications.

Abstract

In this paper, a class of stable explicit $θ$-schemes are proposed for solving anticipated backward stochastic differential equations (anticipated BSDEs) which generator not only contains the present values of the solutions but also the future. We subtly transform the delay process of the generator into the current measurable process, resulting in high-order convergence rate. We also analyze the stability of our numerical schemes and strictly prove the error estimates. Various numerical tests powerful demonstrate high accuracy of the proposed numerical schemes.

Theorems & Definitions (14)

• Remark 2.1
• Remark 2.2
• Lemma 3.1: Discrete Gronwall Lemma
• Lemma 3.2
• Remark 3.3
• Theorem 3.4
• Remark 3.5
• Lemma 3.6
• Theorem 3.7
• Remark 3.8