Viability property for a backward stochastic differential equation and applications to partial differential equations
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.
backward stochastic differential equation
viscosity solutions of semilinear parabolic partial differential equations
scientific article; zbMATH DE number 1501929