Predicting the time at which a L\'evy process attains its ultimate supremum

TL;DR

This paper investigates optimal stopping times for a Lévy process to minimize the L^1-distance to the time of its ultimate supremum, extending previous work from Brownian motion to general Lévy processes with infinite horizon.

Contribution

It generalizes the optimal stopping problem for the ultimate supremum of Lévy processes, providing explicit solutions and conditions based on the process's properties and the law of the supremum.

Findings

No finite L^1-distance stopping time exists if the supremum time has infinite mean.

Optimal stopping strategies depend on the median of the supremum's law, with immediate stopping or threshold-based stopping.

Results are explicitly characterized using scale functions for processes without positive jumps.

Abstract

We consider the problem of finding a stopping time that minimises the $L^1$-distance to $\theta$, the time at which a L\'evy process attains its ultimate supremum. This problem was studied in [12] for a Brownian motion with drift and a finite time horizon. We consider a general L\'evy process and an infinite time horizon (only compound Poisson processes are excluded, furthermore due to the infinite horizon the problem is only interesting when the L\'evy process drifts to $-\infty$). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If $\theta$ has infinite mean there exists no stopping time with a finite $L^1$-distance to $\theta$, whereas if $\theta$ has finite mean it is either optimal to stop immediately or to stop when the process reflected in its supremum exceeds a positive level, depending on whether the median of the law of the ultimate supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the L\'evy process has no positive jumps.