Multidimensional backward stochastic differential equations with uniformly continuous coeffi\-cients
type: Item
instance of: scholarly article
title: Multidimensional backward stochastic differential equations with uniformly continuous coeffi\-cients
zbMATH Open document ID: 1048.60047
published in: Bernoulli
publication date: 2004-06-18T00:00:00Z
review text: Given an \(m\)-dimensional Brownian motion \(W=(W_t)_{t\in [0,1]}\) and a square-integrable \(d\)-dimen\-sional random variable \(\xi\), measurable with respect to the \(\sigma\)-field generated by \(W\), the author studies the existence and the uniqueness of the solution for the backward stochastic differential equation (for short: BSDE) \(dY_t=-f(t,Y_t,Z_t)dt+Z_tdW_t\), \(t\in[0,1]\), \(Y_1=\xi\). Nonlinear BSDEs of this general type were introduced by \textit{E. Pardoux} and \textit{S. G. Peng} [Syst. Control Lett. 14, No. 1, 55--61 (1990; Zbl 0692.93064)]. Since then, motivated by the multitude of applications, many authors have worked on this subject and given improvements of the initial Lipschitz assumptions made by Pardoux and Peng on the driving coefficient \(f\). While in the one-dimensional case (\(d=1\)) the comparison theorem for BSDEs can be used to show that one has the existence of a solution under only continuity of the driver \(f\) [\textit{J.-P. Lepeltier} and \textit{J. San Martín}, Stochastics Stochastics Rep. 63, No. 3/4, 227--240 (1998; Zbl 0910.60046)], one does not have such a comparison result in higher dimensions. Among the recent works generalizing the assumptions on \(f\) in higher dimensions, one should cite that of \textit{S. Hamadène, J.-P. Lepeltier} and \textit{S. Peng} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 111--128 (1997; Zbl 0892.60062)] where the randomness stems from a driving diffusion process, and also the more recent paper by \textit{K. Bahlali, E. H. Essaky, M. Hassani} and \textit{E. Pardoux} [C. R., Math., Acad. Sci. Paris 335, No. 9, 757--762 (2002; Zbl 1017.60068)], who showed the existence, uniqueness and stability of the solutions for a continuous, locally monotone coefficient \(f\) that must not be locally Lipschitz in the variable \(y\) or in the variable \(z\). The author of the present paper, submitted in 2000, supposes that \((y,z)\rightarrow f(t,\omega,y,z)\) is uniform…
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 2075980
zbMATH Keywords: backward stochastic differential equation
zbMATH Keywords: non Lipschitz assumptions
author: Said Hamadène
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3150/bj/1065444816
OpenAlex ID: W2041903514
DOI: 10.3150/BJ/1065444816
Recommended article: \(L^{p}\) solutions of BSDEs with stochastic Lipschitz condition
Recommended article: Q4528681
Recommended article: The uniqueness of solutions of perturbed backward stochastic differential equations
Recommended article: Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution
Recommended article: Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient
Recommended article: A generalized comparison theorem for BSDEs and its applications
Recommended article: \(L^p\) solutions of BSDEs with a new kind of non-Lipschitz coefficients
Recommended article: Backward stochastic differential equations with non-Lipschitz coefficients
Recommended article: Existence and uniqueness of solutions for BSDEs with locally Lipschitz coefficient
Recommended article: On comparison theorem and solutions of BSDEs for Lévy processess
description: scientific article; zbMATH DE number 2075980