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.

We deal with the risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional di erential equations. Using backward stochastic di erential equations we show the existence of an optimal control and, a saddle-point and an equilibrium point for respectively the zero-sum and nonzero-sum games.

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