Wiener disorder problem with observation control

Abstract

Suppose that a Wiener process gains a known drift rate at some unobservable disorder time with some zero-modified exponential distribution. The process is observed only at some intervals that we control. Beginning and end points and the lengths of the observation intervals are controlled optimally. We pay cost for observing the process and for switching on the observation. We show that Bayes optimal alarm times minimizing the expected total cost of false alarms, detection delay cost and observation costs exist. Optimal alarms may occur during the observations or between the observation times when the odds-ratio process hits a set. We derive the sufficient conditions for the existence of the optimal stopping and switching rules and describe the numerical methods to calculate optimal value function.