Predicting the supremum: optimality of "stop at once or not at all"

TL;DR

This paper investigates optimal stopping rules for stochastic processes to maximize the likelihood of stopping near the process's supremum, revealing that the optimal strategy is often to stop either immediately or at the end, depending on process characteristics.

Contribution

It establishes a general 'bang-bang' optimal stopping rule for a broad class of processes, including Lévy processes, under convex reward functions, extending previous results.

Findings

Optimal stopping times are either at the start or end of the process.

Conditions are identified under which the 'stop at once or not at all' rule applies.

Results apply to processes with finite and certain infinite Lévy measures.

Abstract

Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of M_T-X_tau. Under fairly general conditions on the process X_t, it is shown that the optimal stopping time tau is of "bang-bang" form: it is either optimal to stop at time 0 or at time T. For the case of random walk, the rule tau=T is optimal if the steps of the walk stochastically dominate their opposites, and the rule tau=0 is optimal if the reverse relationship holds. For Le'vy processes X_t with finite Le'vy measure, an analogous result is proved assuming that the jumps of X_t satisfy the above condition, and the drift of X_t has the same sign as the mean jump. Finally, conditions are given under which the result can be extended to the case of nonfinite Le'vy measure.