Viscosity solutions of system of PDEs with interconnected obstacles and nonlinear Neumann boundary conditions

This paper investigates the Hamilton-Jacobi-Bellman system of equations associated with the m-states optimal switching problem in finite horizon when the state process is constrained to live in a connected bounded closed domain. We show existence and uniqueness of the solution in viscosity sense of the system. The main tool is the notion of systems of generalized reflected backward stochastic differential equations with oblique reflection and the Feynman-Kac representation of their solutions in the Markovian framework.