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

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.