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Optimally Stopping at a Given Distance from the Ultimate Supremum of a Spectrally Negative Lévy Process

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  17 March 2021

Mónica B. Carvajal Pinto  and
Kees van Schaik Open the ORCID record for Kees van Schaik [Opens in a new window]
Mónica B. Carvajal Pinto*
Affiliation:
University of Manchester
Kees van Schaik*
Affiliation:
University of Manchester
*
*Postal address: Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK. Email address: [email protected]
*Postal address: Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK. Email address: [email protected]
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Abstract

We consider the optimal prediction problem of stopping a spectrally negative Lévy 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.

Keywords

Spectrally negative Lévy processoptimal predictionoptimal stoppingscale functions

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62M20: Prediction
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 279 - 299
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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