On semi-linear degenerate backward stochastic partial differential equations
type: Item
instance of: scholarly article
title: On semi-linear degenerate backward stochastic partial differential equations
zbMATH Open document ID: 1011.60046
author: Ying Hu
author: Jin Ma
author: Jiongmin Yong
publication date: 2002-12-01T00:00:00Z
review text: Given a one-dimensional Brownian motion \(W\), the authors consider the one-dimensional backward stochastic PDE of the form \[ du=-\bigl\{ Lu+Mq+f(t,x,u) \bigr\}dt+ qdW_t,\quad (t,x)\in[0,T] \times R,\;u(T,.)=g, \] where \[ Lu={1\over 2} a(t,x)^2 u_{xx}+ b(t,x)u_x+c(t,x)u,\quad Mq=\alpha (t,x)q_x+ \gamma (t,x)q, \] and \(a,b,c,\alpha, \gamma:[0,T] \times R\times \Omega\to R\), \(g: \Omega\to R\), \(f:[0,T] \times R^2\to R\) satisfy appropriate adaptedness and regularity assumptions. Under the parabolicty condition \(a(t,x)^2-\alpha(t,x)^2 \geq 0\), for all \((t,x)\), the authors study existence and uniqueness of a solution formed by random fields \(u,q\) such that, for each fixed \(x\in R\), \(u(., x,.), q(.,x,.)\) are \(({\mathcal F}^W_t)\)-adapted processes satisfying the above equation in a certain sense. Special cases of this type of equation have been studied by many authors, among them Pardoux (1979), Bensoussan (1982), Hu, Peng (1991). One remarks also that this type of equation contains a special case of nonlinear backward stochastic differential equations \((L=0\), \(M=0\), \(f\) can depend on \((t,x,u)\) but also on \(q)\) introduced by Pardoux and Peng in its general form in 1990. While former works on the topic of such backward stochastic PDEs supposed the so-called superparabolicity condition \(a(t,x)^2-\alpha(t,x)^2 \geq\delta >0\), for all \((t,x)\), \textit{J. Ma} and \textit{J. Yong} [Stochastic Processes Appl. 70, No. 1, 59--84 (1997; Zbl 0911.60048) and Probab. Theory Relat. Fields 113, No. 2, 135--170 (1999; Zbl 0922.60053)] have removed this condition for a class of linear equations satisfying some compatibility assumption on the coefficients (which in particular contains the one-dimensional case with \(f(t,x,u)\equiv h(t,x))\). In the present paper, by establishing some new a priori estimates for both linear and semilinear backward stochastic PDEs, the authors show that regularity and uniform boundedness of adapted solutions can be determined b…
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 1836948
zbMATH Keywords: degenerate backward stochastic partial differential equations
zbMATH Keywords: adapted solutions
zbMATH Keywords: nonlinear Feynman-Kac formula
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: s004400100193
OpenAlex ID: W4255499372
DOI: 10.1007/S004400100193
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1836948