In this paper we study the problem of optimal multiple mode switching in finite horizon when switching a system from a regime i to another one j incurs a payment \(-g_{ij}\) which is not necessarily negative, i.e., switching the controlled system could incur a subsidy. Under the monotonicity and the triangle inequality properties of the switching payments \(g_{ij}\) , we show that the problem is well-posed and we exhibit the optimal strategy. We use probabilistic tools relying on the notion of Snell envelope of processes and systems of reflected backward stochastic differential equations with inter-connected obstacles. At the end of the paper, in the Markovian framework, we show that the vector of value functions of the problem is a unique viscosity solution to its associated HJB system of variational inequalities (or PDEs) with inter-connected obstacles.
