Probabilistic interpretation of a system of semilinear parabolic partial differential equations
type: Item
instance of: scholarly article
title: Probabilistic interpretation of a system of semilinear parabolic partial differential equations
zbMATH Open document ID: 0891.60054
DOI: 10.1016/S0246-0203(97)80101-X
author: Etienne Pardoux
author: Frédéric Pradeilles
author: Zusheng Rao
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1998-07-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77578
review text: The authors introduce a class of backward stochastic differential equations with respect to both a Brownian motion and a finite sequence of Poisson processes. As in the work by \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)] the authors prove the formula \(Z_t^{t,x,n} =\partial Y_t^{t,x,n} \sigma (x,n)\) relating the components \(Z\) and \(Y\) of the solution of the BSDE. As an application the authors provide a stochastic interpretation for the viscosity solution of a system of nonlinear parabolic partial differential equations. The functions \(u_i(t,x) =Y_t^{t,x,i}\), \(1\leq i\leq k\), satisfy \[ {\partial u_i\over\partial t}+ L^iu_i+f_i \bigl(t,x,u(t,x),(\nabla u_i \sigma_i)(t,x)\bigr)=0, \quad u_i(T,x)= g_i(x). \] The uniqueness of the viscosity solution to this equation is proved.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1059907
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: parabolic partial differential equations
zbMATH Keywords: viscosity solutions
reviewed by: David Nualart
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1059907