A Finite Horizon Optimal Multiple Switching Problem

We consider the problem of optimal multiple switching in finite horizon, when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and completely solved using probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov diffusion process, we show that the vector of value functions of the optimal problem is a viscosity solution to a system of variational inequalities with inter-connected obstacles.