Optimal Multiple Stopping with Sum-Payoff

Abstract

We consider the optimal stopping of independent, discrete time sequences $X_1,\dots, X_n$, 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, we derive approximatively optimal stopping times and stopping values. The solutions are obtained via a system of $m$ differential equations of first order. As an application we consider the case that $X_i=c_iZ_i+d_i$, with $(Z_i)$ independent identically distributed in the domain of attraction of an extreme value distribution. We obtain explicit results for stopping values and approximative optimal stopping rules.