Electronic Communications in Probability

A Controller And A Stopper Game With Degenerate Variance Control

Ananda Weerasinghe

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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.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 9, 89-99.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

stochastic games optimal stopping degenerate diffusions saddle point

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Weerasinghe, Ananda. A Controller And A Stopper Game With Degenerate Variance Control. Electron. Commun. Probab. 11 (2006), paper no. 9, 89--99. doi:10.1214/ECP.v11-1202. https://projecteuclid.org/euclid.ecp/1465058852

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