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Backward stochastic differential equations and integral-partial differential equations. (English) Zbl 0878.60036
Backward nonlinear differential equations (BSDE’s) with respect to Brownian motion have been introduced by E. Pardoux and S. G. Peng [Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)]. In a following paper [in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079)] they noted that these BSDE’s provide a probabilistic formula for the solutions of certain classes of systems of quasilinear parabolic partial differential equations (PDE’s) of second order, and are naturally associated with viscosity solutions of such PDE’s. This paper generalizes the above results to the case of BSDE’s with respect to both independent Brownian motion and Poisson random measure. The associate system of parabolic PDE’s is then a system of integro-partial differential equations. Under smooth assumptions the BSDE is proved to have a unique solution. Then in a Markovian framework, some function defined from the solution of the BSDE is the unique viscosity solution of the system of integro-partial differential equations.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
35K55 Nonlinear parabolic equations
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