Naslov
The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

Autori
Avram, Florin ; Grahovac, Danijel ; Vardar-Acar, Ceren

Izvornik
ESAIM-Probability and Statistics (1292-8100) 24 (2020); 454-525

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
spectrally negative processes ; scale functions ; Skorokhod regulation ; capital injections ; dividend optimization

Sažetak
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 Lévy “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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA