The controller-and-stopper game for a linear diffusion

Consider a process X(·) = {X(t), 0 t < 1} with values in the interval I = (0,1), absorption at the boundary-points of I, and dynamics dX(t) = (t)dt + (t)dW(t), X(0) = x. The values ( (t), (t)) are selected by a controller from a subset of < ◊(0,1) that depends on the current position X(t), for every t 0. At any stopping rule of his choice, a second player, called stopper, can halt the evolution of the process X(·), upon which he receives from the controller the amount e u(X( )); here 2 [0,1) is a discount factor, and u : [0,1] ! < is a continuous “reward function”. Under appropriate conditions on this function and on the contoller’s set of choices, it is shown that the two players have a saddle-point 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 inf X(·) E e u(X( )) = inf X(·) sup E e u(X( )) .