Hostname: page-component-6bf8c574d5-m789k Total loading time: 0 Render date: 2025-03-11T17:06:57.673Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix representations of Wiener–Hopf factorisations for Lévy processes

Part of: Stochastic processes Probability theory and 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: [email protected]
**Postal address: Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

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)
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 16
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.Google Scholar
Asmussen, S. (2014). Lévy processes, phase-type distributions and martingales. Stoch. Models 30, 443–468.CrossRefGoogle Scholar
Asmussen, S. (2022). On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance. Finance Stoch. 26, 383–416.CrossRefGoogle Scholar
Asmussen, S. (2022). Remarks on Lévy process simulation. In Advances in Modeling and Simulation. Festschrift for Pierre L’Ecuyer, eds Z. Botev, A. Keller, C. Lemieux, B. Tuffin, Springer, New York, pp. 21–40.CrossRefGoogle Scholar
Asmussen, S. (2024). Erlangization/Canadization of phase-type jump diffusions, with applications to barrier options. J. Indian Soc. Statist. Prob. 25, 575–596.CrossRefGoogle Scholar
Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
Asmussen, S. and Ivanovs, J. (2018). A factorization identity for a Lévy process over a phase-type horizon. Stoch. Models 34, 397–408.CrossRefGoogle Scholar
Avram, F., Pistorius, M. R. and Usabel, M. (2003). The two barriers ruin problem via a Wiener–Hopf decomposition approach. Ann. Univ. Craiova Math. Comp. Sci. Ser. 30, 38–44.Google Scholar
Bladt, M. and Ivanovs, J. (2021). Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon. Stoch. Process. Appl. 142, 105–123.CrossRefGoogle Scholar
Bladt, M. and Nielsen, B. F. (2017). Matrix-Exponential Distributions in Applied Probability. Springer, New York.CrossRefGoogle Scholar
Boxma, O. and Mandjes, M. (2023). A decomposition for Lévy processes sampled at Poisson moments. J. Appl. Prob. 60, 557–569.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. J. Business 75, 305–332.CrossRefGoogle Scholar
Deelstra, G. and Simon, M. (2017). Multivariate European option pricing in a Markov-modulated Lévy framework. J. Comput. Appl. Math. 317, 171–187.CrossRefGoogle Scholar
Hackmann, D. (2022). Wiener–Hopf factorization for the normal inverse Gaussian process. Preprint, arXiv:1902.08804v2.Google Scholar
Hackmann, D. and Kuznetsov, A. (2016). Approximating Lévy processes with completely monotone jumps. Ann. Appl. Prob. 26, 328–359.CrossRefGoogle Scholar
Ivanovs, J. (2021). On scale functions for Lévy processes with negative phase-type jumps. Queueing Systems 98, 3–19.CrossRefGoogle Scholar
Jeannin, M. and Pistorius, M. (2010). A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes. Quant. Finance 10, 629–644.CrossRefGoogle Scholar
Jiang, Z. and Pistorius, M. R. (2008). On perpetual American put valuation and first passage in a regime-switching model with jumps. Finance Stoch. 12, 331–355.CrossRefGoogle Scholar
Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Prob. 20, 1801–1830.CrossRefGoogle Scholar
Kuznetsov, A. (2010). Wiener–Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Prob. 47, 1023–1033.CrossRefGoogle Scholar
Kuznetsov, A. (2012). On the distribution of exponential functionals for Lévy processes with jumps of rational transform. Stoch. Process. Appl. 122, 654–663.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J.-C. and Van Schaik, K. (2011). A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Prob. 21, 2171–2190.CrossRefGoogle Scholar
Kuznetsov, A. and Peng, X. (2012). On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps. Stoch. Process. Appl. 122, 2610–2638.CrossRefGoogle Scholar
Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf for Lévy processes having positive jumps with rational transform. J. Appl. Prob. 45, 118–134.CrossRefGoogle Scholar
Mordecki, E. (2002). The distribution of the maximum of a Lévy process with positive jumps of phase-type. In Proc. Conf. Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko, Kyiv, pp. 309–316.Google Scholar
Mordecki, E. (2003). Ruin probabilities for Lévy processes with mixed-exponential negative jumps. Teor. Veroyat Primen. 48, 188–194.Google Scholar
Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy processes. J. Appl. Prob. 43, 208–220.CrossRefGoogle Scholar
Weisstein, E. W. Padé approximant. From MathWorld – A Wolfram web resource. https://mathworld.wolfram.com/PadeApproximant.html.Google Scholar