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

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.