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

Khaled Bahlali (1) , Lucian Maticiuc (2) , Adrian Zalinescu (3)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 Faculty of Mathematics, "Alexandru Ioan Cuza" University, Iaşi
3 Alexandru Ioan Cuza" University of Iasi and "Octav Mayer"Mathematics Institute of the Romanian Academy, Romania

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

Keywords: Backward stochastic differential equations; Reflecting stochastic differential equation; Penalization method; Weak solution; Jakubowski S-topology; Backward stochastic differential equations.

Domains: Mathematics [math]; Probability [math.PR]

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