# Article

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Keywords:
discrete-time Markov process; risk-seeking expected total cost; optimal stopping rule; stability index; total variation metric
Summary:
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 $p(\cdot |x)$, $x\in X$ is approximated by the transition probability $\widetilde{p}(\cdot |x)$, $x\in X$, and that the stopping rule $\widetilde{f}_*$ , which is optimal for the process with the transition probability $\widetilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: $\sup_{x\in X}\|p(\cdot |x)-\widetilde{p}(\cdot |x)\|)$ for an additional cost paid for using the rule $\widetilde{f}_*$ instead of the (unknown) stopping rule $f_*$ optimal for $p$.
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