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

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  • M'onica B. Carvajal Pinto
  • Kees van Schaik

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.

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  • M'onica B. Carvajal Pinto & Kees van Schaik, 2019. "Optimally stopping at a given distance from the ultimate supremum of a spectrally negative L\'evy process," Papers 1904.11911, arXiv.org, revised Jul 2020.
    • Handle: RePEc:arx:papers:1904.11911
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    References listed on IDEAS

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    1. Cohen, Albert, 2010. "Examples of optimal prediction in the infinite horizon case," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 950-957, June.
    2. L. Alili & A. E. Kyprianou, 2005. "Some remarks on first passage of Levy processes, the American put and pasting principles," Papers math/0508487, arXiv.org.
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