On Asymptotics of Optimal Stopping Times

Abstract

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Asymptotic Duration for Optimal Multiple Stopping Problems

“…Differential equations were also used in [18,19]. In this study, we extend some of our results in [1] to the multiple stopping case through some inductive arguments and verify these results with simulations. For simplicity, we only analyse continuous distributions and reserve the notation f (x), F(x), and h(y) = 1 − F(x) for the continuous probability density, cumulative distribution, and "survivor" function, respectively.…”

“…For the first stop, we would arrive on y 2 = 0.6987, since y 3 is the first variable to satisfy 1 and for the second, we arrive on y 5 = 0.8968, as this is the first subsequent variable for which y m 2 ≥ v N−m 2 ,1 . This particular example would have resulted in the reward 0.6987 + 0.8968 = 1.5955.…”

“…, y m . In [1], we used the pay-off of y m (i.e., the value of the variable stopped on). To extend this to the multiple optimal stopping problem we stop on k > 1 variables and, after stopping on variables y m 1 , y m 2 , .…”

“…In [1], we outlined a novel approach for a general asymptotic technique for calculating the asymptotic behaviour of the pay-off as well as E(τ N ) and Var(τ N ) in the single-stopping case, where τ N is the single-stopping time, as N → ∞ for general classes of probability distributions in the full-information problem where we wish to maximise the expected reward y m . The techniques in our previous paper, which are extended in this study, employ the asymptotic analysis of difference and differential equations in order to establish and solve asymptotic differential equations for the quantities of interest.…”

A new method for computing asymptotic results in optimal stopping problems

“…Solving differential equations in order to determine asymptotic values in optimal stopping problems and, more specifically, in variants of the secretary problem has numerous precedents [5,6,7,11,12,18,19,20,24,27]. This is not surprising, given the relationship between difference and differential equations.…”