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