TheW,Zscale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems
| Autor: | Danijel Grahovac, Ceren Vardar-Acar, Florin Avram |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Skorokhod regulation dividend optimization Spectrally negative processes Markov process Scale (descriptive set theory) 01 natural sciences Lévy process 010104 statistics & probability symbols.namesake Absorption (logic) [MATH]Mathematics [math] 0101 mathematics processes with Poissonian/Parisian observations Mathematics Discrete mathematics Gerber-Shiu functions Mathematical finance capital injections 010102 general mathematics spectrally negative processes scale functions Harmonic function generalized drawdown stopping symbols Interval (graph theory) Variety (universal algebra) |
| Zdroj: | ESAIM: Probability and Statistics ESAIM: Probability and Statistics, EDP Sciences, 2020, 24, pp.454-525. ⟨10.1051/ps/2019022⟩ |
| ISSN: | 1262-3318 1292-8100 |
| DOI: | 10.1051/ps/2019022 |
| Popis: | 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)WandZ. 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,Zalphabet”. It is important to note – as is currently being shown – that these identities apply equally to other spectrally negative Markov processes, where however theW,Zfunctions are typically much harder to compute. We collect below our favorite recipes from the “W,Zkit”, 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. |
| Databáze: | OpenAIRE |
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