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About stability of risk-seeking optimal stopping. (English) Zbl 1300.60059
Summary: We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chains 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 \(\widetilde{f}_{\ast}\), 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 the additional cost paid for using the rule \(\widetilde{f}_{\ast}\) instead of the (unknown) stopping rule \(f_{\ast}\) optimal for \(p\).
60G40 Stopping times; optimal stopping problems; gambling theory
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