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Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition. (English) Zbl 1217.60047

The authors study a class of generalized reflected backward stochastic differential equations (GRBSDEs) driven by the Teugels martingales associated with a Lévy process as well as by a square integrable increasing process; the reflecting lower barrier is supposed to be square integrable and càdlàg. The present work generalizes earlier results of E. Pardoux and S. Zhang [Probab. Theory Relat. Fields 110, No. 4, 535–558 (1998; Zbl 0909.60046)], M. El Otmani [J. Appl. Math. Stochastic Anal. 2006, Article ID 85407 (2006; Zbl 1147.60319)] and N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez [Ann. Probab. 25, No. 2, 702–737 (1997; Zbl 0899.60047)] by bringing together the concepts of generalized backward stochastic differential equations and reflected backward stochastic differential equations. Let us also mention that Y. Ren and L. Hu [Stat. Probab. Lett. 77, No. 15, 1559–1566 (2007; Zbl 1128.60048)] studied reflected backward stochastic differential equations driven by Teugels martingales and an independent Brownian motion. By combining the methods developed in these four papers, the authors of the present work prove the existence and the uniqueness of the solution. For this, they use the Snell envelope, the penalization method and a fixed point theorem. Furthermore, they prove a comparison theorem for the solutions of GRBSDEs. In the end, they give in terms of a forward equation and an associated GRBSDE the probabilistic interpretation for the viscosity solution of an obstacle problem for partial differential-integral equations with a nonlinear Neumann boundary condition.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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