Optimality of the one step look-ahead stopping times

Abstract

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn , n = 0, 1, …}; X 0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X 1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex [αNg(XN )I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.