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A controller and a stopper game with degenerate variance control. (English) Zbl 1119.91018
Summary: We consider a zero sum stochastic differential game which involves two players, the controller and the stopper. The stopper selects the stopping rule which halts the game. The controller chooses the diffusion coefficient of the corresponding state process which is allowed to degenerate. At the end of the game, the controller pays the stopper, the amount $E{\int }_{0}^{\tau }{e}^{-\alpha t}C\left({Z}_{x}\left(t\right)\right)dt$, where ${Z}_{x}\left(·\right)$ represents the state process with initial position $x$ and $\alpha$ is a positive constant. Here $C\left(·\right)$ is a reward function where the set $x:C\left(x\right)>0$ is an open interval which contains the origin. Under some assumptions on the reward function $C\left(·\right)$ and the drift coefficient of the state process, we show that this game has a value. Furthermore, this value function is Lipschitz continuous, but it fails to be a ${C}^{1}$ function.
##### MSC:
 91A23 Differential games (game theory) 60G40 Stopping times; optimal stopping problems; gambling theory 91A15 Stochastic games 93E20 Optimal stochastic control (systems)