On semi-linear degenerate backward stochastic partial differential equations
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 by those of the coefficients, a special feature that comes from the ``backward character'' of the equation. The well-posedness and the regularity of adapted solutions is then proved by a combination of a priori estimates for both linear and semilinear equations and a so-called bootstrap method. As applications the authors prove some comparison results for adapted solutions of backward stochastic PDEs and generalize a Feynman-Kac formula developed 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)].
degenerate backward stochastic partial differential equations
scientific article; zbMATH DE number 1836948