A QUICKEST DETECTION PROBLEM WITH AN OBSERVATION COST

• 成果类型: Article
• 署名作者: Dalang, Robert C.; Shiryaev, Albert N.
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1028
• 发表日期: 2015
• 页码: 1475-1512

摘要

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift changes into a Brownian motion with positive drift. The change occurs at an unknown disorder time with exponential distribution. There is a penalty for declaring too early that the change has occurred, and a cost for late detection proportional to the time between occurrence of the change and the time when the change is declared. Here, we consider the case where there is also a cost for observing the process. This stochastic control problem can be formulated using either the notion of strong solution or of weak solution of the s.d.e. that defines the observation process. We show that the value function is the same in both cases, even though no optimal strategy exists in the strong formulation. We determine the optimal strategy in the weak formulation and show, using a form of the principle of smooth fit and under natural hypotheses on the parameters of the problem, that the optimal strategy takes the form of a two-threshold policy: observe only when the posterior probability that the change has already occurred, given the observations, is larger than a threshold A >= 0, and declare that the disorder time has occurred when this posterior probability exceeds a threshold B >= A. The constants A and B are determined explicitly from the parameters of the problem.