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.
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DOI : 10.1051/ps/2019022
Mots-clés : Spectrally negative processes, scale functions, Gerber-Shiu functions, Skorokhod regulation, dividend optimization, capital injections, processes with Poissonian/Parisian observations, generalized drawdown stopping
@article{PS_2020__24_1_454_0,
author = {Avram, Florin and Grahovac, Danijel and Vardar-Acar, Ceren},
title = {The {\protect\emph{W},} {\protect\emph{Z}} scale functions kit for first passage problems of spectrally negative {L\'evy} processes, and applications to control problems},
journal = {ESAIM: Probability and Statistics},
pages = {454--525},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2019022},
mrnumber = {4158667},
zbl = {1461.60028},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2019022/}
}
TY - JOUR AU - Avram, Florin AU - Grahovac, Danijel AU - Vardar-Acar, Ceren TI - The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems JO - ESAIM: Probability and Statistics PY - 2020 SP - 454 EP - 525 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2019022/ DO - 10.1051/ps/2019022 LA - en ID - PS_2020__24_1_454_0 ER -
%0 Journal Article %A Avram, Florin %A Grahovac, Danijel %A Vardar-Acar, Ceren %T The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems %J ESAIM: Probability and Statistics %D 2020 %P 454-525 %V 24 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2019022/ %R 10.1051/ps/2019022 %G en %F PS_2020__24_1_454_0
Avram, Florin; Grahovac, Danijel; Vardar-Acar, Ceren. The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 454-525. doi : 10.1051/ps/2019022. http://www.numdam.org/articles/10.1051/ps/2019022/
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