A linear numerical scheme for nonlinear BSDEs with uniformly continuous coefficients

Omid S Fard,Ali V Kamyad

doi:10.1155/s1110757x04401168

Abstract

We attempt to present a new numerical approach to solve nonlinear backward stochastic differential equations. First, we present some definitions and theorems to obtain the condition, from which we can approximate the nonlinear term of the backward stochastic differential equation (BSDE) and we get a continuous piecewise linear BSDE corresponding to the original BSDE. We use the relationship between backward stochastic differential equations and stochastic controls by interpreting BSDEs as some stochastic optimal control problems to solve the approximated BSDE and we prove that the approximated solution converges to the exact solution of the original nonlinear BSDE.

Highlights

The backward stochastic differential equation (BSDE) theory and its applications have been a focus of interest in recent years
X(T) = ξ, where ξ is a random variable that will become certain only at the terminal time T. This type of equation, at least in nonlinear case, was first introduced by Pardoux and Peng [14], who proved the existence and uniqueness of a solution under suitable assumptions on f and ξ. Their aim was to give a probabilistic interpretation of a solution to a second-order quasilinear partial differential equation
The Black-Scholes formula for option pricing can be recovered via a system of forward-backward stochastic differential equations (FBSDEs)

Summary

Introduction

The backward stochastic differential equation (BSDE) theory and its applications have been a focus of interest in recent years. In the Markovian case, Douglas et al [7] established a numerical method for a class of forward-backward SDEs—a more general version of the BSDEs (1.1)—based on a fourstep scheme developed by Ma et al [11]. Chevance, in his Ph.D. thesis [6], proposed a numerical method for BSDEs by using binomial approach to approximate the process x. Zhang, in his Ph.D. thesis [16], presented a numerical method for a class of BSDEs, whose terminal value ξ takes the form φ(x), where x is a diffusion process, and φ(·) is a so-called L∞-Lipschitz functional. Afterwards, we will find the optimal control problem associated with the obtained piecewise linear BSDE

Problem formulation and preliminaries

Continuous piecewise linear approximation

Convergence

Metamorphosis

Numerical results