scholarly journals The Theory of Scale Functions for Spectrally Negative Lévy Processes

2012 ◽  
pp. 97-186 ◽  
Author(s):  
Alexey Kuznetsov ◽  
Andreas E. Kyprianou ◽  
Victor Rivero
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Scale Functions ◽  
Spectrally Negative Lévy Processes

scholarly journals Smoothness of scale functions for spectrally negative Lévy processes

2010 ◽  
Vol 150 (3-4) ◽  
pp. 691-708 ◽  
Author(s):  
T. Chan ◽  
A. E. Kyprianou ◽  
M. Savov
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Scale Functions ◽  
Spectrally Negative Lévy Processes

scholarly journals Exit problems for general draw-down times of spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 441-457 ◽  
Author(s):  
Bo Li ◽  
Nhat Linh Vu ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Scale Functions ◽  
Dynamic Level ◽  
Running Maximum ◽  
Potential Measure ◽  
Exit Problems ◽  
Spectrally Negative Lévy Processes

AbstractFor spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

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

2020 ◽  
Vol 24 ◽  
pp. 454-525 ◽  
Author(s):  
Florin Avram ◽  
Danijel Grahovac ◽  
Ceren Vardar-Acar
Keyword(s):  
Lévy Processes ◽  
Mathematical Finance ◽  
Levy Processes ◽  
Small Sample ◽  
First Passage ◽  
Scale Functions ◽  
Exit Problems ◽  
Finance Risk ◽  
Storage Theory ◽  
Spectrally Negative Lévy Processes

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.

scholarly journals On the refracted–reflected spectrally negative Lévy processes

2018 ◽  
Vol 128 (1) ◽  
pp. 306-331 ◽  
Author(s):  
José-Luis Pérez ◽  
Kazutoshi Yamazaki
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Spectrally Negative Lévy Processes
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