Backward stochastic differential equations associated to a symmetric Markov process
type: Item
instance of: scholarly article
title: Backward stochastic differential equations associated to a symmetric Markov process
zbMATH Open document ID: 1071.60067
published in: Potential Analysis
publication date: 2005-05-13T00:00:00Z
full work available at URL: RR-4425.pdf
review text: The authors consider a semi-linear symmetric second order differential operator \(L\) (in divergence form, without non degeneracy assumption but with some closability condition) having measurable coefficients, and they study the system of PDE's \[ (\partial_{t} +L)u(t,x)+f(t,x,u,\nabla u\sigma)=0,\;(t,x)\in[0,T]\times R^{d},\quad u(T,x)=\Phi(x),\tag{1} \] to which they associate backward stochastic differential equations (BSDE). If \(f\) and the coefficients of \(L\) are Lipschitz continuous, (1) has a viscosity solution \(u\); a diffusion process \(X_{s}^{t,x}\), \(s\in[ t,T],(t,x)\in[0,T]\times R^{d}\), with infinitesimal operator \(L\) and driven by some Brownian motion \(B\) can be defined, and a stochastic interpretation of \(u\) can be given with the help of the solution \(Y_{s}^{t,x}=u(s,X_{s}^{t,x})\), \(Z_{s}^{t,x}=(\nabla u\sigma)(s,X_{s}^{t,x})\) of the associated BSDE [\textit{E. Pardoux} and \textit{S. Peng}, Syst. Control Lett. 14, No.~1, 55--61 (1990; Zbl 0692.93064) and in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200--217 (1992; Zbl 0766.60079)]. Further studies about the connection between BSDE and PDE were done by \textit{G. Barles} and \textit{E. Lesigne} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 47--80 (1997; Zbl 0886.60049)] and \textit{V. Bally} and \textit{A. Matoussi} [J. Theor. Probab. 14, No.~1, 125--164 (2001; Zbl 0982.60057)] (for solutions \(u\) in Sobolev spaces and associated BSDE). For measurable coefficients of \(L\), \textit{A. Lejay} [Stochastic Processes Appl. 97, No.~1, 1--39 (2002; Zbl 1058.60045)] studied BSDE's associated to PDE's, which are driven by a Dirichlet process. In the case of only measurable coefficients, the symmetric Markov process \(X\) generated by \(L\) does no more satisfy an SDE. The authors replace the usual stochastic calculus corresponding to SDE's by Fukushima's stochastic calculus (1994). By using functi…
reviewed by: Rainer Buckdahn
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K45
zbMATH DE Number: 2168219
zbMATH Keywords: divergence form semilinear partial differential equations
author: Vlad Bally
author: Etienne Pardoux
author: Lucretiu Stoica
MaRDI profile type: Publication
OpenAlex ID: W2056932936
cites work: Weak solutions for SPDE's and backward doubly stochastic differential equations
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cites work: \(L^p\) solutions of backward stochastic differential equations.
cites work: Dirichlet forms and symmetric Markov processes
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cites work: Backward Stochastic Differential Equations in Finance
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DOI: 10.1007/S11118-003-6457-8
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description: scientific article; zbMATH DE number 2168219