Abstract
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 to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational 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.
Highlights
IntroductionController–stopper games are two-player stochastic dynamic games, whose payoffs depend on the evolution over time of some state variable, one player can control its
Controller–stopper games are two-player stochastic dynamic games, whose payoffs depend on the evolution over time of some state variable, one player can control its Journal of Optimization Theory and Applications (2020) 186:688–724 dynamics, while the other player can stop the game
We mention Karatzas and Sudderth [4], who derived the explicit solution for a game with a one-dimensional diffusion with absorption at the endpoints of a bounded interval as a state process; Karatzas and Zamfirescu [5,6] developed a martingale approach to a general class of controller–stopper games, while Bayraktar and Huang [7] showed that the value functions of such games are the unique viscosity solution to an appropriate Hamilton–Jacobi–Bellman equation
Summary
Controller–stopper games are two-player stochastic dynamic games, whose payoffs depend on the evolution over time of some state variable, one player can control its. We consider the case of a controller facing fixed and proportional costs every time he moves the state variable, so that intervening continuously over time is clearly not feasible for him In this context, the controller will make use of impulse controls, which are sequences of interventions times and corresponding intervention sizes, describing when and by how much will the controlled process be shifted. Impulse controls have been studied in stochastic differential games as well, and as in the controller–stopper case, most of the research has been done in the zero-sum framework For this reason, it is worth mentioning the work by Aïd et al [20], who developed a general model for nonzero sum impulse games implementing a verification theorem which provides an appropriate system of quasivariational inequalities for the equilibrium payoffs and related strategies of the two players. Some numerical experiments illustrate the qualitative behaviour of such equilibria
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