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Stochastic Optimal Multi-Modes Switching with a Viscosity Solution Approach

We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary ($g_{ij}(t,x)\geq 0$). We show existence of the optimal strategy, and give when the optimal strategy is finite via a verification theorem. Finally, when the state of the system is a markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of $m$ variational partial differential inequalities with inter-connected obstacles.