Article2006

Generalized Reflected BSDE and an Obstacle Problem for PDEs with a Nonlinear Neumann Boundary Condition

49 citations0/0 files storedStochastic Analysis and Applications

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Differential Equations and Boundary Problems

Abstract

In this article, we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equation involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary, in using the penalization method. We also give a characterization of the solution as the value function of an optimal stopping time problem. Then we give a probabilistic formula for the viscosity solution of an obstacle problem for PDEs with a nonlinear Neumann boundary condition.

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References

1990Adapted solution of a backward stochastic differential equation 1984Stochastic differential equations with reflecting boundary conditions 2005Backward stochastic differential equations and quasilinear parabolic partial differential equations 1997Reflected solutions of backward SDE's, and related obstacle problems for PDE's 1991Probabilistic interpretation for systems of quasilinear parabolic partial differential equations 1996Backward stochastic differential equations with reflection and Dynkin games 1998Generalized BSDEs and nonlinear Neumann boundary value problems 2005Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier

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2009On Solutions of Backward Stochastic Volterra Integral Equations with Jumps in Hilbert Spaces 2009Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition 2008Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes

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