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 solutions of penalized partial differential equations
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K. Bahlali, Lucian Maticiuc, Adrian Zalinescu. Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (102), pp.1-19. ⟨10.1214/EJP.v18-2467⟩. ⟨hal-00998298⟩

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