Characterization of optimality in classes of “truncatable” stopping rules

Abstract

Let \(X_1,X_2,\ldots ,X_n,\ldots \) be a discrete-time stochastic process. The following optimal stopping problem is considered. We observe \(X_1,X_2,\ldots \) on the one-by-one basis getting successively the data \(x_1,x_2,\ldots \). At each stage \(n\), \(n=1,2,\ldots \), after the data \(x_1,\ldots ,x_n\) have been observed, we may stop, and if we stop, we gain \(g_n(x_1,\ldots ,x_n)\). In this article, we characterize the structure of all optimal (randomized) stopping times \(\tau \) that maximize the average gain value \(G(\tau )=E g_\tau (X_1,\ldots , X_\tau )\) in some natural classes of stopping times \(\tau \) we call truncatable: \(\tau \) is called truncatable if \(G(\tau \wedge N)\rightarrow G(\tau )\) as \(N\rightarrow \infty \). It is shown that under some additional conditions on the structure of \(g_n\) (suitable for statistical applications) every finite (with probability 1) stopping time is truncatable.