Suppose that the process $X=\{X_n,n\in \mathbb{N}\}$ is observed sequentially. There are two random moments of time ${\theta }_1$ and ${\theta }_2$, independent of X, and X is a Markov process given ${\theta }_1$ and ${\theta }_2$. The transition probabilities of X change for the first time at time ${\theta }_1$ and for the second time at time ${\theta }_2$. 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...

We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chain on a Borel space $X$. It is supposed that the transition probability $p(·|x)$, $x\in X$ is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and that the stopping rule ${\tilde{f}}_{*}$ , which is optimal for the process with the transition probability $\tilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: ${sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel )$ for...