Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T10:06:28.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Alan L. Lewis  and
Ernesto Mordecki
Alan L. Lewis*
Affiliation:
optioncity.net
Ernesto Mordecki*
Affiliation:
Universidad de la República
*
∗Postal address: 983 Bayside Cove, Newport Beach, CA 92660, USA. Email address: alewis@optioncity.net
∗∗Postal address: Facultad de Ciencias, Centro de Matemática, Universidad de la República, Iguá 4225, CP 11400 Montevideo, Uruguay. Email address: mordecki@cmat.edu.uy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

Keywords

Lévy processWiener-Hopf factorizationBaxter-Donsker formularuin probabilityrational Laplace transformladder process

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60J50: Boundary theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 118 - 134
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Asmussen, S. (1977). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2) World Scientific, Singapore.Google Scholar
[2] Asmussen, S., Avram, F. and Pistorius, M. (2004). Russian and American put options under exponential phase-type Lévy motion. Stoch. Process. Appl. 109, 79–111.Google Scholar
[3] Baxter, G. and Donsker, M. D. (1957). On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85, 73–87.CrossRefGoogle Scholar
[4] Bertoin, J. (1994). Lévy processes that can creep downwards never increase. Ann. Inst. H. Poincaré Prob. Statist. 31, 379–391.Google Scholar
[5] Bertoin, J. (1996). Lévy Processes (Cambridge Tracts Math. 121). Cambridge University Press.Google Scholar
[6] Bertoin, J. and Doney, R. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
[7] Boyarchenko, S. I. and Levendorskiı˘, S. Z. (2002). Non-Gaussian Merton–Black–Scholes Theory (Adv. Ser. Statist. Sci. Appl. Prob. 9) World Scientific, Singapore.Google Scholar
[8] Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance Rev. Assoc. Française Finance 25, 71–94.Google Scholar
[9] Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566–576.Google Scholar
[10] Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
[11] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1, 59–74.Google Scholar
[12] Harrison, J. M. (1977). The supremum distribution of a Lévy process with no negative Jumps. Adv. Appl. Prob. 9, 417–422.CrossRefGoogle Scholar
[13] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.CrossRefGoogle Scholar
[14] Kemperman, J. (1961). The Passage Problem for a Stationary Markov Chain. University of Chicago Press.Google Scholar
[15] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
[16] Korevaar, J. (2002). A century of complex Tauberian theory. Bull. Amer. Math. Soc. (N.S.) 39, 475–531.Google Scholar
[17] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[18] Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 4, 473–493.Google Scholar
[19] Mordecki, E. (2002). The distribution of the maximum of a Lévy process with positive Jumps of phase-type. Theory Stoch. Process. 8, 309–316.Google Scholar
[20] Mordecki, E. (2003). Ruin probabilities for Lévy processes with mixed-exponential negative Jumps. Theory Prob. Appl. 48, 170–176.Google Scholar
[21] Petrov, V. (1987). Limit Theorems for Sums of Independent Random Variables. Nauka, Moscow (in Russian).Google Scholar
[22] Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy process. J. Appl. Prob. 43, 208–220.Google Scholar
[23] Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 21–34.Google Scholar
[24] Rogozin, B. (1966). On distributions of functionals related to boundary problems for processes with independent increments. Theory Prob. Appl. 11, 580–591.Google Scholar
[25] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies Adv. Math. 68). Cambridge University Press.Google Scholar
[26] Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 65, 243–256.Google Scholar
[27] Zolotarev, V. M. (1964). The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–662.CrossRefGoogle Scholar