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Markov chain approximations to scale functions of Lévy processes

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Mijatović, Aleksandar, Vidmar, Matija and Jacka, Saul D. (2015) Markov chain approximations to scale functions of Lévy processes. Stochastic Processes and their Applications, 125 (10). pp. 3932-3957. doi:10.1016/j.spa.2015.05.012

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Official URL: http://dx.doi.org/10.1016/j.spa.2015.05.012

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Abstract

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the Lévy triplet of X. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Markov processes, Lévy processes, Brownian motion processes, Semimartingales (Mathematics)
Journal or Publication Title: Stochastic Processes and their Applications
Publisher: Elsevier Science BV
ISSN: 0304-4149
Official Date: October 2015
Dates:
DateEvent
October 2015Published
28 May 2014Available
17 May 2015Accepted
3 December 2014Submitted
Volume: 125
Number: 10
Page Range: pp. 3932-3957
DOI: 10.1016/j.spa.2015.05.012
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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