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∈X is approximated by the transition probability p˜(⋅|x), x∈X, and that the stopping rule f˜∗ , which is optimal for the process with the transition probability p˜ is applied to the process with the transition probability p. We give an upper bound (expressed in term of the total variation distance: supx∈X∥p(⋅|x)−p˜(⋅|x)∥) for an additional cost paid for using the rule f˜∗ instead of the (unknown) stopping rule f∗ optimal for p.
In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. The problem is to minimize an average sample number given that Bayesian risk due to incorrect decision does not exceed some given bound. We characterize the form of optimal sequential stopping rules in this problem. In particular, we have a characterization of the form of optimal sequential decision procedures when the Bayesian risk includes both the loss due to incorrect decision and the cost of observations.
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∈X, is approximated by the transition probability p˜(⋅|x), x∈X, and the stopping rule τ˜∗, optimal for 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 \wtτ∗, and the minimal total expected cost. The bound given is a constant times \dpssupx∈X∥p(⋅|x)−\wtp(⋅|x)∥, where ∥⋅∥is the total variation norm.