Volume 24, 2020
|Page(s)||454 - 525|
|Published online||06 October 2020|
The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems
Laboratoire de Mathématiques Appliquées, Université de Pau, France.
2 Department of Mathematics, University of Osijek, Croatia.
3 Department of Statistics, Middle East Technical University, Ankara, Turkey.
*Corresponding author: firstname.lastname@example.org
Accepted: 6 November 2019
In the last years there appeared a great variety of identities for first passage problems of spectrally negative Lévy processes, which can all be expressed in terms of two “q-harmonic functions” (or scale functions) W and Z. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental “two-sided exit” problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the “W, Z alphabet”. It is important to note – as is currently being shown – that these identities apply equally to other spectrally negative Markov processes, where however the W, Z functions are typically much harder to compute. We collect below our favorite recipes from the “W, Z kit”, drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.
Mathematics Subject Classification: 60G51 / 60G40 / 60J45
Key words: Spectrally negative processes / scale functions / Gerber-Shiu functions / Skorokhod regulation / dividend optimization / capital injections / processes with Poissonian/Parisian observations / generalized drawdown stopping
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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