Viability property for a backward stochastic differential equation and applications to partial differential equations
type: Item
instance of: scholarly article
title: Viability property for a backward stochastic differential equation and applications to partial differential equations
zbMATH Open document ID: 0969.60061
author: Rainer Buckdahn
author: Marc Quincampoix
author: Aurel Răşcanu
publication date: 2001-01-30T00:00:00Z
review text: Given a backward stochastic differential equation [see \textit{E. Pardoux} and \textit{S. Peng}, Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)], one can ask which are the conditions such that the solution remains in a given set of constraints. The aim of the paper is to give some necessary and sufficient conditions for this question. The authors apply then the result to the study of a system of semilinear parabolic partial differential equations whose (viscosity) solutions can be expressed [see also \textit{E. Pardoux} and \textit{S. Peng}, in: Stochastic partial differential equations and their applications Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079) or \textit{S. Peng}, Stochastics Stochastics Rep. 37, No. 1/2, 61-74 (1991; Zbl 0739.60060)] in terms of solution of a suitable backward stochastic differential equation. The paper is based on the study of the viability property of solutions [see \textit{J.-P. Aubin}, ``Viability theory'' (1991; Zbl 0755.93003) for partial differential equations and \textit{J.-P. Aubin} and \textit{G. Da Prato}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 4, 595-613 (1990; Zbl 0741.60046), Stochastic Anal. Appl. 13, No. 1, 1-11 (1995; Zbl 0816.60053) and ibid. 16, No. 1, 1-15 (1998; Zbl 0931.60059) for stochastic differential equations]. In the present paper the approach is based on the convexity and on the Taylor expansion of the distance function of the closed convex set of constraints.
reviewed by: Mihai Gradinaru
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1501929
zbMATH Keywords: backward stochastic differential equation
zbMATH Keywords: viability property
zbMATH Keywords: viscosity solutions of semilinear parabolic partial differential equations
MaRDI profile type: Publication
DOI: 10.1007/S004409900038
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1501929