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

Published online by Cambridge University Press:  17 March 2021

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]

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.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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