Optimal multiple stopping with sum-payoff (Q2845217)

The paper deals with the optimal multiple stopping problem for a sequence of independent random variables \(X_1,\dots,X_n\), where \(m\) stops are allowed. The payoff is the sum of the stopped values. Under the assumption of convergence of related imbedded point processes to a Poisson process in the plane, the approximate optimal stopping times and stopping values are obtained via a system of \(m\) differential equations of first order. The result are applied to the case that \(X_i = c_iZ_i +d_i\), with \(\{Z_i\}_{i=1}^n\) independent identically distributed in the domain of attraction of an extreme value distribution. The explicit results for stopping values and approximate optimal stopping rules are derived. The results are based on those of \textit{R. Kühne} and \textit{L. Rüschendorf} [Theory Probab. Appl. 48, No. 3, 465--480 (2003) and Teor. Veroyatn. Primen. 48, No. 3, 557--575 (2003; Zbl 1079.60043)] and of the authors [Adv. Appl. Probab. 43, No. 4, 1086--1108 (2011; Zbl 1235.60038); Ann. Appl. Probab. 21, No. 5, 1965--1993 (2011; Zbl 1251.60038)].