CHARACTERIZATIONS OF OPTIMAL POLICIES IN A GENERAL STOPPING PROBLEM AND STABILITY ESTIMATING

We consider an optimal stopping problem for a general discrete-time process X 1 , X 2 , …, X n , … on a common measurable space. Stopping at time n ( n = 1, 2, …) yields a reward R n ( X 1 , …, X n ) ≥ 0, while if we do not stop, we pay c n ( X 1 , …, X n ) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain: $$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$ We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules. In the particular case of Markov sequence X 1 , X 2 , … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.