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The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems
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Date
2019-11-01
Author
Avram, Florin
Grahovac, Danijel
Vardar Acar, Ceren
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In the last years there appeared a great variety of identities for first passage problems of spectrally negative Levy 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.
Subject Keywords
Statistics and Probability
URI
https://hdl.handle.net/11511/40553
Journal
ESAIM-Probability and Statistics
DOI
https://doi.org/10.1051/ps/2019022
Collections
Department of Statistics, Article
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F. Avram, D. Grahovac, and C. Vardar Acar, “The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems,”
ESAIM-Probability and Statistics
, pp. 1–65, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/40553.