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We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $X$. It is supposed that an unknown transition probability $p(·|x)$, $x\in X$, is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and the stopping rule ${\tilde{\tau }}_{*}$, optimal for $\tilde{p}$, is applied to the process governed by $p$. We found an upper bound for the difference between the total expected cost, resulting when applying ${\tilde{\tau }}_{*}$, and the minimal total expected cost. The bound given is a constant times ${\displaystyle {sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel }$, where $\parallel ·\parallel$ is the total variation...

We study the stability of average optimal control of general discrete-time Markov processes. Under certain ergodicity and Lipschitz conditions the stability index is bounded by a constant times the Prokhorov distance between distributions of random vectors determinating the “original and the perturbated” control processes.

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...

We study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations $X_1,X_2,\cdots$ when testing two simple hypotheses about their common density $f$: $f=f_0$ versus $f=f_1$. As a functional to be minimized, it is used a weighted sum of the average (under $f_0$) sample number and the two types error probabilities. We prove that the problem is reduced to stopping time optimization for a ratio process generated by $X_1,X_2,\cdots$ with the density $f_0$. For ${\tau }_{*}$ being the corresponding...