Optimal Stopping by Means of Point Process Observations with Applications in Reliability

A problem in reliability is considered in which only partial information is available. Some technical system is assumed to work in one of N unobservable states. The changes of the states are driven by a Markov process with known characteristics. The system fails from time to time according to a point process with a failure rate (intensity) which depends on the unobservable state. After failure a minimal repair is carried out immediately which leaves the state of the system unchanged. It is investigated under which conditions there exists an optimal time to stop operating the system with respect to some reward functional. The only available information is given by the failure point process observations. An explicit solution to this optimal stopping problem with partial information is derived. The problem is solved in the martingale framework. Results for monotone stopping problems are used and a generalization of the so-called monotone case is considered. The well-known disruption or disorder or detection ...