# Explicit $\theta$-Schemes for Solving Anticipated Backward Stochastic Differential Equations

## TL;DR

This paper introduces explicit -schemes for solving anticipated backward stochastic differential equations, transforming the delay process to achieve high-order convergence and demonstrating their stability and accuracy through theoretical analysis and numerical tests.

## Contribution

The paper proposes a novel class of explicit -schemes that handle anticipated BSDEs with delays, achieving high-order convergence and stability.

## Key findings

- Schemes are stable and have high-order convergence.
- Numerical tests confirm high accuracy.
- Error estimates are rigorously proved.

## Abstract

In this paper, a class of stable explicit $\theta$-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.