Second-order schemes for solving decoupled forward backward stochastic differential equations

Abstract

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are driven by a ddimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17,25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.Mathematics subject classification: 60H35, 60H10, 65C20, 65C30

Optimal Error Estimates of the Crank-Nicolson Scheme for Solving a Kind of Decoupled FBSDEs

“…In this paper, we study the error estimate of the Crank-Nicolson scheme proposed in [10] for solving a kind of decoupled FBSDEs. Under weaker conditions than that in [9], we rigorously prove the second order convergence rate of the Crank-Nicolson scheme.…”

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

“…In this paper, we considered the theoretical error estimates of the Crank-Nicolson (C-N) scheme for solving decoupled FBSDEs proposed in [29]. By properly using the Young's inequality to the error equations of the C-N scheme and their associated variational equations, we first rigorously obtained a general error estimate result for the C-N scheme.…”

“…Now, based on the reference equations (3.3), (3.4), (3.6) and (3.7), we introduce the Crank-Nicolson scheme (Scheme 2.1 proposed in [29]) for solving decoupled FBSDEs (1.1).…”

“…To our best knowledge in the literature, up to now, one-step second-order numerical schemes for solving BSDE and decoupled FBSDEs were proposed and studied in [5,25,28,29,32]. In 2006, Zhao, Chen and Peng proposed numerical schemes for solving BSDE in [25], in which the Crank-Nicolson (short for C-N) is included.…”

“…It was proposed in [25] for solving BSDE and the extension for solving decoupled FBSDEs was introduced in [29]. The second-order convergence rate of the C-N scheme for BSDE was proved in [28], but for decoupled FBSDEs is still open until now.…”