# Optimally stopping at a given distance from the ultimate supremum of a spectrally negative L\'evy process

## TL;DR

This paper solves an optimal stopping problem for spectrally negative Lévy processes, characterizing the optimal strategy explicitly and revealing a threshold-based structure depending on the distance parameter.

## Contribution

It provides a fully explicit solution to the stopping problem using scale functions, revealing a novel threshold-based decision rule depending on the parameter $b$.

## Key findings

- Optimal stopping rule characterized explicitly.
- Threshold structure depends on the parameter $b$.
- Solution applicable under mild conditions.

## Abstract

We consider the optimal prediction problem of stopping a spectrally negative L\'evy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We find that the solution has an interesting non-trivial structure: if $b$ is larger than a certain threshold then it is optimal to stop as soon as the difference between the running supremum and the position of the process exceeds a certain level (less than $b$), while if $b$ is smaller than this threshold then it is optimal to stop immediately (independent of the running supremum and position of the process). We also present some examples.