Two-Player Nonzero-Sum Stochastic Differential Games with Switching Controls

In this paper, a two-player nonzero-sum stochastic differential game problem is studied with both players using switching controls. A verification theorem associated with a set of variational inequalities is established as a sufficient criterion for Nash equilibrium, in which the equilibrium switching strategies for the two players, indicating when and where it is optimal to switch, are characterized in terms of the so-called switching regions and continuation regions. The verification theorem is proved in a piecewise way along the sequence of total decision times of the two players. Then, some detailed explanations are also provided to illustrate the idea why the conditions are imposed in the verification theorem.