Explicit High Order One-Step Methods for Decoupled Forward Backward Stochastic Differential Equations

Abstract

By using the Feynman-Kac formula and combining with It ô-Taylor expansion and finite difference approximation, we first develop an explicit third order onestep method for solving decoupled forward backward stochastic differential equations. Then based on the third order one, an explicit fourth order method is further proposed. Several numerical tests are also presented to illustrate the stability and high order accuracy of the proposed methods.

High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

“…After that, by combining the Itô-Taylor expansion and finite difference approximations for the first-order and second-order derivatives, the authors in [40] proposed an explicit fourth order one-step method for solving general decoupled forward backward stochastic differential equations (FBSDEs), which can be convergent with fourth order when the forward stochastic differential equations (SDEs) are solved by the weak order 4.0 Itô-Taylor scheme.…”

“…Based on the nonlinear Feynman-Kac formula, by using the Itô-Taylor expansions and high order finite difference approximations for derivatives, we give the discretizations of the solutions of the BSDE (1). Then by using the predictioncorrection method based on the fourth order one-step scheme in [40], we propose an explicit fifth order one-step scheme for solving the BSDE (1). Based on the fifth order scheme, we use the prediction-correction method again to further propose an explicit sixth order one-step scheme.…”