Refracted Lévy processes

A. E. Kyprianou; R. L. Loeffen

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 24-44
  • ISSN: 0246-0203

Abstract

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Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1{Ut>b} dt+dXt, where X={Xt : t≥0} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

How to cite

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Kyprianou, A. E., and Loeffen, R. L.. "Refracted Lévy processes." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 24-44. <http://eudml.org/doc/240802>.

@article{Kyprianou2010,
abstract = {Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1\{Ut&gt;b\} dt+dXt, where X=\{Xt : t≥0\} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.},
author = {Kyprianou, A. E., Loeffen, R. L.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic control; fluctuation theory; Lévy processes},
language = {eng},
number = {1},
pages = {24-44},
publisher = {Gauthier-Villars},
title = {Refracted Lévy processes},
url = {http://eudml.org/doc/240802},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Kyprianou, A. E.
AU - Loeffen, R. L.
TI - Refracted Lévy processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 24
EP - 44
AB - Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1{Ut&gt;b} dt+dXt, where X={Xt : t≥0} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.
LA - eng
KW - stochastic control; fluctuation theory; Lévy processes
UR - http://eudml.org/doc/240802
ER -

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