Numerical method for backward stochastic differential equations
type: Item
instance of: scholarly article
title: Numerical method for backward stochastic differential equations
zbMATH Open document ID: 1017.60074
author: Jin Ma
author: Jaime San Martín
author: Soledad Torres
author: Philip Protter
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \(W\) be a \(d\)-dimensional Brownian motion. The authors develop a new method of approximating solutions \(Y\) of the multidimensional backward stochastic differential equation (BSDE) \[ dY_t= -f(t, Y_t)dt+ Z_t dW_t,\quad t\in [0,T], \] with a continuous driver \(f\) which is Lipschtz in the \(y\)-variable and independent of \(z\). As a consequence the interest in numerical treatments of such equations in finance, several numerical approximating methods for BSDEs have already been developed, cf. \textit{J. Douglas jun.}, \textit{J. Ma} and \textit{P. Protter} [Ann. Appl. Probab. 6, No. 3, 940-968 (1996; Zbl 0861.65131)], \textit{V. Bally} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 177-191 (1997; Zbl 0889.60068)], \textit{D. Chevance} [in: Numerical methods in finance, 232-244 (1997; Zbl 0898.90031)] and \textit{V. Bally}, \textit{G. Pagès} and \textit{J. Printems} [Monte Carlo Methods Appl. 7, No. 1/2, 21-33 (2001)]. Another type of approximating BSDEs uses the discretization of the filtration [e.g. \textit{F. Coquet}, \textit{V. Mackevičius} and \textit{J. Mémin}, Stochastic Processes Appl. 75, No. 2, 235-248 (1998; Zbl 0932.60047)]. In their present paper the authors propose a discretization of the above BSDE where the Brownian motion is replaced by a simple random walk. Although the method of the authors uses ideas similar to those used in the above papers, in that they also approximate the Brownian motion by a discrete process, their main feature is that they do not need the convergence of the discretized filtrations to the original Brownian one in order to get the weak convergence in the Skorokhod topology of the solutions. Such a relaxation reduces the complexity in the construction of approximating solutions.
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 65C30
zbMATH DE Number: 1906190
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: numerical methods
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015961165
OpenAlex ID: W2039366144
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DOI: 10.1214/AOAP/1015961165
description: scientific article; zbMATH DE number 1906190