A Markov renewal approach to the Poisson disorder problem

In the Poisson disorder problem, a decision maker observes a Poisson process whose intensity changes at some random time. After the change, the process is considered to be in a state of disorder. The decision maker wishes to use one action before disorder and another, different, action after disorder. However, the change to disorder produces no direct discernible evidence-the decision maker perceives only the events of the Poisson process. The decision to switch from one action to the other must be based exclusively on the evidence provided by these observed events. For many cost functionals, an optimal policy is to change actions when the probability of disorder (given all past observations) rises above a certain level. Considering a certain Markov renewal process embedded in the disorder process yields new information about such policies.