THE UNIQUENESS OF OPTIMAL SEQUENTIAL DECISION

The statistical sequential analysis is actually an optimal-stopping-tme problem~[1].There has been relatively much discussion on the existence of optimal stopping time for an integrable adapted sequence X=(X_n, J_n)_1~∞. If there exists an optimal stopping time, how to characterize it and whether to be unique are the natural questions to ask. The aim of the present paper is to answer these questions by characterizing an optimal stopping time and give criteria to justify the uniqueness of optimal stopping time. For some special adapted sequence, we make further studies on the uniqueness of optimal stopping time, prove that the optimal stopping solution of the famous secretary is unique, and show that Wald's sequential probability ratio test is the unique optimal test when the distribution function of the probability ratio is continuous. The burglar problem is finally studied and a rule of writing out all the optimal stopping times is given.