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

M\'onica B. Carvajal Pinto and Kees van Schaik

Papers from arXiv.org

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.

Date: 2019-04, Revised 2020-07
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