Optimal strategies for discovering new species in continuous time

Abstract

Consider a population of distinct species S_j, j∈J, members of which are selected at different time points T₁, T₂,· ··, one at each time. Assume linear costs per unit of time and that a reward is earned at each discovery epoch of a new species. We treat the problem of finding a selection rule which maximizes the expected payoff. As the times between successive selections are supposed to be continuous random variables, we are dealing with a continuous-time optimal stopping problem which is the natural generalization of the one Rasmussen and Starr (1979) have investigated; namely, the corresponding problem with fixed times between successive selections. However, in contrast to their discrete-time setting the derivation of an optimal strategy appears to be much harder in our model as generally we are no longer in the monotone case.

This note gives a general point process formulation for this problem, leading in particular to an equivalent stopping problem via stochastic intensities which is easier to handle. Then we present a formal derivation of the optimal stopping time under the stronger assumption of i.i.d. (X₁, A₁) (X₂, A₂), · ·· where X_n gives the label (j for S_j) of the species selected at T_n and A_n denotes the time between the nth and (n – 1)th selection, i.e. A_n = T_n – T_n–1. In the case where even X_n and A_n are independent and A_n has an IFR (increasing failure rate) distribution, an explicit solution for the optimal strategy is derived as a simple consequence.