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July 2001 The Controller-and-Stopper Game for a Linear Diffusion
Ioannis Karatzas, William D. Sudderth
Ann. Probab. 29(3): 1111-1127 (July 2001). DOI: 10.1214/aop/1015345598

## Abstract

Consider a process $X(\cdot) = {X(t), 0 \leq t < \infty}$ with values in the interval $I = (0, 1)$, absorption at the boundary points of $I$, and dynamics

$$dX(t) = \beta(t)dt + \sigma(t)dW(t),\quad X(0) = x.$$

The values $(\beta(t), \sigma(t))$ are selected by a controller from a subset of $\Re \times (0, \infty)$ that depends on the current position $X(t)$, for every $t \geq 0$. At any stopping rule $\tau$ of his choice, a second player, called a stopper, can halt the evolution of the process $X(\cdot)$, upon which he receives from the controller the amount $e^{-\alpha\tau}u(X(\tau))$; here $\alpha \epsilon [0, \infty)$ is a discount factor, and $u: [0, 1] \to \Re$ is a continuous “reward function.” Under appropriate conditions on this function and on the controller’s set of choices, it is shown that the two players have a saddlepoint of “optimal strategies.” These can be described fairly explicitly by reduction to a suitable problem of optimal stopping, whose maximal expected reward V coincides with the value of the game,

$$V = \sup_{\tau} \inf_{X(\cdot)} \mathbf{E}[e^{-\alpha\tau}u(X(\tau))] = \inf_{X(\cdot)} \sup_{\tau} \mathbf{E}[e^{-\alpha\tau}u(X(\tau))].$$

## Citation

Ioannis Karatzas. William D. Sudderth. "The Controller-and-Stopper Game for a Linear Diffusion." Ann. Probab. 29 (3) 1111 - 1127, July 2001. https://doi.org/10.1214/aop/1015345598

## Information

Published: July 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1039.60043
MathSciNet: MR1872738
Digital Object Identifier: 10.1214/aop/1015345598

Subjects:
Primary: 60G40 , 93E20
Secondary: 60D60 , 62L15

Keywords: excessive functions , generalized Itô rule , Local time , one-dimensional diffusions , Optimal stopping , stochastic game