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This paper studies a system of $m$ variational inequalities withinterconnected obstacles in infinite horizon associated to optimal multi-modesswitching problems. Our main result is the existence and uniqueness of acontinuous solution in viscosity sense, for that system. The proof of the mainresult strongly relies on the connection between the systems of variationalinequalities and reflected backward stochastic differential equations (RBSDEs)with oblique reflection, which will be characterized through a Feynman-Kac'sformula. The main feature of our system of infinite horizon RBSDEs is that itscomponents are interconnected through both the generators and the obstacles.

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles