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Characterization of optimality in classes of “truncatable” stopping rules. (English) Zbl 1311.62124
Summary: Let $$X_1$$, $$X_2,\dots$$, $$X_n,\dots$$ be a discrete-time stochastic process. The following optimal stopping problem is considered. We observe $$X_1$$, $$X_2,\dots$$ on the one-by-one basis getting successively the data $$x_1$$, $$x_2,\dots$$. At each stage $$n$$, $$n=1,2,\dots$$, after the data $$x_1,\dots,x_n$$ have been observed, we may stop, and if we stop, we gain $$g_n(x_1,\dots,x_n)$$. In this article, we characterize the structure of all optimal (randomized) stopping times $$\tau$$ that maximize the average gain value $$G(\tau )=E g_\tau (X_1,\dots , X_\tau )$$ in some natural classes of stopping times $$\tau$$ we call truncatable: $$\tau$$ is called truncatable if $$G(\tau \wedge N)\to G(\tau )$$ as $$N\to \infty$$. 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.
##### MSC:
 62L10 Sequential statistical analysis 62L15 Optimal stopping in statistics 62C10 Bayesian problems; characterization of Bayes procedures
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