Electronic Journal of Probability

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

Khaled Bahlali, Lucian Maticiuc, and Adrian Zalinescu

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H99: None of the above, but in this section
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations

Reflecting stochastic differential equation Penalization method Weak solution Jakubowski S-topology Backward stochastic differential equations

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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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