Optimality of the one step look-ahead stopping times

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 { X n , 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 A α g ( x ) = αE x g ( X 1 ) – g ( x ) to be the infinitesimal generator of g . If τ is any stopping time satisfying the conditions: E x [ α N g ( X N ) I ( τ > N )] → 0 as as N → ∞ , then Applications of the results are considered.