On the equality of solutions of max-min and min-max systems of variational inequalities with interconnected bilateral obstacles

In this paper, we deal with the solutions of systems of PDEs with bilateral interconnected obstacles of min-max and max-min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min-max and max-min systems coincide. Then, this common viscosity solution is related to a multi-dimensional doubly reflected BSDE with bilateral interconnected obstacles. Finally, its relationship with the values of a zero-sum switching game is studied.