A two-disorder detection problem

Suppose that the process ${\displaystyle X=\{X_n,n\in N\}}$ is observed sequentially. There are two random moments of time ${\displaystyle {\theta }_1}$ and ${\displaystyle {\theta }_2}$, independent of X, and X is a Markov process given ${\displaystyle {\theta }_1}$ and ${\displaystyle {\theta }_2}$. The transition probabilities of X change for the first time at time ${\displaystyle {\theta }_1}$ and for the second time at time ${\displaystyle {\theta }_2}$. Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy is found and the corresponding maximal probability is calculated.