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In this paper we study the optimal m-states switching problem in finitehorizon as well as infinite horizon with risk of default. We allow theswitching cost functionals and cost of default to be of polynomial growth andarbitrary. We show uniqueness of a solution for a system of m variationalpartial differential inequalities with inter-connected obstacles. This systemis the deterministic version of the Verification Theorem of the Markovianoptimal m-states switching problem with risk of default. This problem isconnected with the valuation of a power plant in the energy market.

Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default