Robustness Estimating of Optimal Stopping Problem with Unbounded Revenue and Cost Functions

We study the stability of the optimal stopping problem for a discrete-time Markov process on a general space state X. Revenue and cost functions are allowed to be unbounded. The stability (robustness) is understood in the sense that an unknown transition probability p(·|x), x ∈ X, is approximated by the known one p̃(·|x), x ∈ X, and the stopping rule τ̃∗, optimal for the process governed by p̃ is applied to the original process represented by p. The criteria of stopping rule optimization is the total expected return. We give an upper bound for the decrease of the return due to the replacement of the unknown optimal stopping rule τ∗ by its approximation τ̃∗. The bound is expressed in terms of the weighted total variation distance between the transition probabilities p and p̃. AMS Subject Classification: 60G40, 90C40