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2006 A Controller And A Stopper Game With Degenerate Variance Control
Ananda Weerasinghe
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Electron. Commun. Probab. 11: 89-99 (2006). DOI: 10.1214/ECP.v11-1202


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(Z_x(t))dt $, where $Z_x(\cdot)$ represents the state process with initial position $x$ and $\alpha $ is a positive constant. Here $C(\cdot)$ is a reward function where the set $\{x: C(x) > 0\}$ is an open interval which contains the origin. Under some assumptions on the reward function $C(\cdot)$ 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.


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Ananda Weerasinghe. "A Controller And A Stopper Game With Degenerate Variance Control." Electron. Commun. Probab. 11 89 - 99, 2006.


Accepted: 4 July 2006; Published: 2006
First available in Project Euclid: 4 June 2016

zbMATH: 1119.91018
MathSciNet: MR2231736
Digital Object Identifier: 10.1214/ECP.v11-1202

Primary: 93E20
Secondary: 60G40

Keywords: Degenerate diffusions , Optimal stopping , saddle point , Stochastic games

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