The Annals of Applied Probability

A Wiener–Hopf Monte Carlo simulation technique for Lévy processes

A. Kuznetsov, A. E. Kyprianou, J. C. Pardo, and K. van Schaik

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Abstract

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s so-called “Canadization” technique as well as Doney’s method of stochastic bounds for Lévy processes; see Carr [Rev. Fin. Studies 11 (1998) 597–626] and Doney [Ann. Probab. 32 (2004) 1545–1552]. We rely fundamentally on the Wiener–Hopf decomposition for Lévy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Lévy en titillant la factorization de Wiener–Hopf (2002) Laboratoire de Mathématiques de L’INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801–1830]. We illustrate our Wiener–Hopf Monte Carlo method on a number of different processes, including a new family of Lévy processes called hypergeometric Lévy processes. Moreover, we illustrate the robustness of working with a Wiener–Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Lévy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 6 (2011), 2171-2190.

Dates
First available in Project Euclid: 23 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1322057319

Digital Object Identifier
doi:10.1214/10-AAP746

Mathematical Reviews number (MathSciNet)
MR2895413

Zentralblatt MATH identifier
1245.65005

Subjects
Primary: 65C05: Monte Carlo methods 68U20: Simulation [See also 65Cxx]

Keywords
Lévy processes exotic option pricing Wiener–Hopf factorization

Citation

Kuznetsov, A.; Kyprianou, A. E.; Pardo, J. C.; van Schaik, K. A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab. 21 (2011), no. 6, 2171--2190. doi:10.1214/10-AAP746. https://projecteuclid.org/euclid.aoap/1322057319


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