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2013 Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs
Khaled Bahlali, Lucian Maticiuc, Adrian Zalinescu
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Electron. J. Probab. 18: 1-19 (2013). DOI: 10.1214/EJP.v18-2467


In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by the penalized partial differential equation.


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Khaled Bahlali. Lucian Maticiuc. Adrian Zalinescu. "Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs." Electron. J. Probab. 18 1 - 19, 2013.


Accepted: 27 November 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1292.35344
MathSciNet: MR3145049
Digital Object Identifier: 10.1214/EJP.v18-2467

Primary: 60H99
Secondary: 35K61 , 60H30

Keywords: Backward stochastic differential equations , Jakubowski S-topology , penalization method , Reflecting stochastic differential equation , Weak solution

Vol.18 • 2013
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