On the optimal stopping values induced by general dependence structures (Q2774444)

The problem of optimal stopping of a sequence \({\mathbf X}=(X_1,\ldots,X_n)\) of \(n\) random variables with distribution functions \(F_1,\ldots,F_n\) is investigated. Denote by \(V({\mathbf X})=\sup_{\tau\in {\mathcal T}} EX_\tau\) the value of \(\mathbf X\), where \(\mathcal T\) is the class of stopping times with respect to \({\mathcal F}_k= \sigma\{X_1,\ldots,X_k\}\), \(k=1,2,\ldots,n\). \(V({\mathbf X})\) depends on the marginals of \(\mathbf X\) and their dependence structure. \textit{Y.~Rinott} and \textit{E.~Samuel-Cahn} [Ann. Stat. 15, 1482-1490 (1987; Zbl 0639.60052) and J. Multivariate Anal. 37, No. 1, 104-114 (1991; Zbl 0722.60040)] have shown that a weak condition of negative dependence leads to an increase of the optimal stopping value compared to the case of independent components with the same marginals. The maximal and minimal values of the optimal stopping problem are determined within the class of all joint distributions with fixed marginals. They correspond to some sort of strong negative or positive dependence of the random variables. The minimal value is determined based on lattice properties of stochastic orderings. A special construction of a Snell envelope which is based on an extension of \textit{V. Strassen}'s theorem on convex domination is used [Ann. Math. Stat. 36, 423-439 (1965; Zbl 0135.18701)].