Optimal on-line detection of outside observations

Suppose that a Markov process X₁,X₂,… appears consecutively. There are two random moments of time θ₁ and θ₂, independent of (X_n). The distribution of the process (X_n) changes for the first time at instant θ₁ and for the second time at instant θ₂. Our objective is to find a stopping rule based only upon the former values of (X_n) which maximizes the probability that the moment of stopping is between θ₁ and θ₂. A sufficient condition is given under which an optimal stopping time is finite. The maximal probability corresponding to the optimal rule is found.