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Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. Kuznetsov
A. Kuznetsov*
Affiliation:
York University
*
∗Postal address: Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada. Email address: kuznetsov@mathstat.yorku.ca
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Abstract

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In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

Keywords

Lévy processWiener-Hopf factorizationmeromorphic functioninfinite producttheta function

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1023 - 1033
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
[2] 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
[3] Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distribution and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.CrossRefGoogle Scholar
[4] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[5] Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Prob. 43, 967–983.CrossRefGoogle Scholar
[6] Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stoch. Process. Appl. 119, 980–1000.CrossRefGoogle Scholar
[7] Čhebotarev, N. G. and Meiman, N. N. (1949). The Routh-Hurwitz problem for polynomials and entire functions. Real quasipolynomials with r=3, s=1. Trudy Mat. Inst. Steklov 26, 3–331 (in Russian).Google Scholar
[8] Doney, R. A. (1987). On Wiener–Hopf factorization and the distribution of extrema for certain stable processes. Ann. Prob. 15, 1352–1362.CrossRefGoogle Scholar
[9] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes (Lecture Notes Math. 1897). Springer, Berlin.Google Scholar
[10] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series and Products, 7th edn. Academic Press, San Diego, CA.Google Scholar
[11] Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Prob. 20, 1801–1830.CrossRefGoogle Scholar
[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[13] Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Prob. 20, 522–564.CrossRefGoogle Scholar
[14] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth 22, 205–225.CrossRefGoogle Scholar
[15] Levin, B. Ya. (1996). Lectures on Entire Functions (Trans. Math. Monogr. 150). American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
[16] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive Jumps with rational transforms. J. Appl. Prob. 45, 118–134.CrossRefGoogle Scholar
[17] Madan, D. B. and Seneta, E. (1990). The Variance Gamma (VG) model for share market returns. J. Business 63, 511–524.CrossRefGoogle Scholar
[18] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar