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.
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 study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations X1,X2,… when testing two simple hypotheses about their common density f: f=f0 versus f=f1. As a functional to be minimized, it is used a weighted sum of the average (under f0) 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 X1,X2,… with the density f0. For τ∗ being the corresponding optimal stopping time we consider a situation when this rule is applied for testing between f0 and an alternative f~1, where f~1 is some approximation to f1. An inequality is obtained which gives an upper bound for the expected cost excess, when τ∗ is used instead of the rule τ~∗ optimal for the pair (f0,f~1). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs (f0,f1) and (f0,f~1).
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.