  We study a two-player nonzero-sum stochastic differential game where one
player controls the state variable via additive impulses while the other player
can stop the game at any time. The main goal of this work is characterize Nash
equilibria through a verification theorem, which identifies a new system of
quasi-variational inequalities whose solution gives equilibrium payoffs with
the correspondent strategies. Moreover, we apply the verification theorem to a
game with a one-dimensional state variable, evolving as a scaled Brownian
motion, and with linear payoff and costs for both players. Two types of Nash
equilibrium are fully characterized, i.e. semi-explicit expressions for the
equilibrium strategies and associated payoffs are provided. Both equilibria are
of threshold type: in one equilibrium players' intervention are not
simultaneous, while in the other one the first player induces her competitor to
stop the game. Finally, we provide some numerical results describing the
qualitative properties of both types of equilibrium.

Probability;Optimization and Control