Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

Abstract

Let (X t )0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (X t ) ‘as close as possible’ to its eventual supremum M T := sup0 ≤ t ≤ T X t , when the reward for stopping at time τ ≤ T is a nonincreasing convex function of M T - X τ. Under fairly general conditions on the process (X t ), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.