The controller-and-stopper game for a linear diffusion

• 成果类型: Article
• 署名作者: Karatzas, I; Sudderth, WD
• 署名单位: Columbia University; Columbia University; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2001
• 页码: 1111-1127
• 关键词: time; goal

摘要

Consider a process X(.) = {X(t), 0 less than or equal to t less than or equal to infinity) 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), X(0) = x. The values beta(t), sigma(t)) are selected by a controller from a subset of 91 x (0, infinity) that depends on the current position X(t), for every t greater than or equal to 0. At any stopping rule tau of his choice, a second player, called a stopper, can halt the evolution of the process X(.), upon which he receives from the controller the amount e(-alphatau) u(X(tau)); here alpha is an element of [0, infinity) is a discount factor, and u: [0, 1] --> R 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, [GRAPHICS]