Fluctuation theory for level-dependent Lévy risk processes

Irmina Czarna, José-Luis Pérez, Tomasz Rolski and Kazutoshi Yamazaki

Stochastic Processes and their Applications, 2019, vol. 129, issue 12, 5406-5449

Abstract: A level-dependent Lévy process solves the stochastic differential equation dU(t)=dX(t)−ϕ(U(t))dt, where X is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ϕk(x)=∑j=1kδj1{x≥bj}. A general rate function ϕ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for U can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.

Keywords: Refracted Lévy process; Multi-refracted Lévy process; Level-dependent Lévy process; Lévy process; Volterra equation; Fluctuation theory (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414919301413
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:129:y:2019:i:12:p:5406-5449

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2019.03.006

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().