A numerical scheme for BSDEs
type: Item
instance of: scholarly article
title: A numerical scheme for BSDEs
zbMATH Open document ID: 1056.60067
author: Jianfeng Zhang
published in: The Annals of Applied Probability
publication date: 2004-06-10T00:00:00Z
review text: The author considers the decoupled system of forward-backward stochastic differential equations \[ \begin{aligned} X_t&=x + \int_0^t b(s,X_s) \,ds + \int_0^t \sigma(s,X_s)\,dW_s,\\ Y_t&= \Phi(X)+\int_t^Tf(s,X_s,Y_s,Z_s)\,ds-\int_t^TZ_s\,dW_s, \end{aligned} \tag{1} \] where \(b,\,\sigma\) and \(f\) are deterministic functions and \(W\) is a standard Brownian motion. The deterministic functional \(\Phi\) is assumed to be a so-called \(L^\infty\)-Lipschitz or a \(L^1\)-Lipschitz functional or takes the form \(g(X_T)\). Based on results of \textit{J. Ma} and \textit{J. Zhang} [Probab. Theory Relat. Fields 122, No. 2, 163--190 (2002; Zbl 1014.60060)] he proves \(L^2\)-regularity of the martingale integrand \(Z\). A numerical scheme is proposed, consisting of the Euler method for the forward diffusion \(X\) and an approximation via step processes for the pair \((Y,Z)\). Convergence in the mean-square sense of the method is proved and convergence rates are obtained. The latter depend in particular on the above stated smoothness properties of \(\Phi\).
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 2072705
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: Euler method
zbMATH Keywords: convergence in mean-square
zbMATH Keywords: \(L^2\) regularity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1075828058
OpenAlex ID: W2095445700
cites work: Discretization error in simulation of one-dimensional reflecting Brownian motion
cites work: Arbitrage Theory in Continuous Time
cites work: Donsker-type theorem for BSDEs
cites work: Numerical methods for forward-backward stochastic differential equations
cites work: Q4344073
cites work: Backward Stochastic Differential Equations in Finance
cites work: Q5653395
cites work: Q4039796
cites work: Q4004325
cites work: Numerical method for backward stochastic differential equations
cites work: Solving forward-backward stochastic differential equations explicitly -- a four step scheme
cites work: Forward-backward stochastic differential equations and their applications
cites work: Representation theorems for backward stochastic differential equations
cites work: Path regularity for solutions of backward stochastic differential equations
cites work: Q4842684
cites work: Adapted solution of a backward stochastic differential equation
cites work: Q4029028
cites work: Discretizing a backward stochastic differential equation
DOI: 10.1214/AOAP/1075828058
description: scientific article; zbMATH DE number 2072705