Optimal stopping rule for the full-information duration problem with random horizon

The full-information duration problem with a random number N of objects is considered. These objects appear sequentially and their values Xk are observed, where Xk, independent of N, are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the kth object is a relative maximum if Xk = max{X1, X2, . . ., Xk}). We assume that N is a random variable with a known upper bound n, so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is N or n. The structure of the optimal rule, which depends on the prior distribution assumed on N, is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that N / n converges, as n → ∞, to a random variable Vm having density fVm(v) = m(1 - v)m-1, 0 ≤ v ≤ 1 for a positive integer m. An interesting feature is that the optimal duration (relative to n) for Model 2 is just (m + 1) times as large as that for Model 1 asymptotically.