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Matrix representations of Wiener–Hopf factorisations for Lévy processes

Part of: Probability theory and stochastic processes Stochastic processes

Published online by Cambridge University Press:  05 March 2025

Søren Asmussen  and
Mogens Bladt Open the ORCID record for Mogens Bladt [Opens in a new window]
Søren Asmussen*
Affiliation:
Aarhus University
Mogens Bladt*
Affiliation:
University of Copenhagen
*
*Postal address: Department of Mathematics, Ny Munkegade, 8000 Aarhus C, Denmark. Email: asmus@math.au.dk
**Postal address: Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. Email: bladt@math.ku.dk
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Abstract

The Wiener–Hopf factors of a Lévy process are the maximum and the displacement from the maximum at an independent exponential time. The majority of explicit solutions assume the upward jumps to be either phase-type or to have a rational Laplace transform, in which case the traditional expressions are lengthy expansions in terms of roots located by means of Rouché’s theorem. As an alternative, compact matrix formulas are derived, with parameters computable by iteration schemes.

Keywords

Phase-type distributionmatrix-exponential distributionrational Laplace transformiterative schemeCGMY processPadé approximationbarrier optionsPoisson sampling

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60-08: Computational methods (not classified at a more specific level)

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 3 , September 2025 , pp. 1028 - 1043
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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