System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

In this paper, we consider a system of fully non linear second order parabolic partial differential equations with interconnected obstacles and boundary conditions on non smooth time-dependent domains. We prove existence and uniqueness of a continuous viscosity solution. This system is the HJB system of equations associated with a m-switching problem in finite horizon, when the state process is the solution of an obliquely reflected stochastic differential equation in non smooth time-dependent domain. Our approach is based on the study of related system of reflected generalized backward stochastic differential equations with oblique reflection. We show that this system has a unique solution which is the optimal payoff and provides the optimal strategy for the switching problem. Methods of the theory of generalized BSDEs and their connection with PDEs with boundary condition are then used to give a probabilistic representation for the solution of the PDE system.