Nonzero-sum stochastic differential games with impulse controls : a verification theorem with applications

Abstract

We consider a general nonzero-sum impulse game with two players. The main\nmathematical contribution of the paper is a verification theorem which\nprovides, under some regularity conditions, a suitable system of\nquasi-variational inequalities for the value functions and the optimal\nstrategies of the two players. As an application, we study an impulse game with\na one-dimensional state variable, following a real-valued scaled Brownian\nmotion, and two players with linear and symmetric running payoffs. We fully\ncharacterize a Nash equilibrium and provide explicit expressions for the\noptimal strategies and the value functions. We also prove some asymptotic\nresults with respect to the intervention costs. Finally, we consider two\nfurther non-symmetric examples where a Nash equilibrium is found numerically.\n