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
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