## Electronic Journal of Probability

### Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 102, 19 pp.

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

https://projecteuclid.org/euclid.ejp/1465064327

Digital Object Identifier
doi:10.1214/EJP.v18-2467

Mathematical Reviews number (MathSciNet)
MR3145049

Zentralblatt MATH identifier
1292.35344

Rights

#### Citation

Bahlali, Khaled; Maticiuc, Lucian; Zalinescu, Adrian. Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs. Electron. J. Probab. 18 (2013), paper no. 102, 19 pp. doi:10.1214/EJP.v18-2467. https://projecteuclid.org/euclid.ejp/1465064327

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