Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms

Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms

Alan L. Lewis and Ernesto Mordecki

Source: J. Appl. Probab. Volume 45, Number 1 (2008), 118-134.

Abstract

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.

First Page: Show

Primary Subjects: 60G51

Secondary Subjects: 60J50

Keywords: Lévy process; Wiener-Hopf factorization; Baxter-Donsker formula; ruin probability; rational Laplace transform; ladder process

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jap/1208358956
Digital Object Identifier: doi:10.1239/jap/1208358956
Mathematical Reviews number (MathSciNet): MR2409315

back to Table of Contents

References

Asmussen, S. (1977). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2) World Scientific, Singapore.

Mathematical Reviews (MathSciNet): MR1794582

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.

Mathematical Reviews (MathSciNet): MR2024845

Digital Object Identifier: doi:10.1016/j.spa.2003.07.005

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.

Mathematical Reviews (MathSciNet): MR84900

Digital Object Identifier: doi:10.2307/1992962

JSTOR: links.jstor.org

Zentralblatt MATH: 0078.32002

Bertoin, J. (1994). Lévy processes that can creep downwards never increase. Ann. Inst. H. Poincaré Prob. Statist. 31, 379--391.

Mathematical Reviews (MathSciNet): MR1324813

Bertoin, J. (1996). Lévy Processes (Cambridge Tracts Math. 121). Cambridge University Press.

Mathematical Reviews (MathSciNet): MR1406564

Bertoin, J. and Doney, R. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363--365.

Mathematical Reviews (MathSciNet): MR1325211

Boyarchenko, S. I. and Levendorskiĭ, S. Z. (2002). Non-Gaussian Merton--Black--Scholes Theory (Adv. Ser. Statist. Sci. Appl. Prob. 9) World Scientific, Singapore.

Mathematical Reviews (MathSciNet): MR1904936

Zentralblatt MATH: 0997.91031

Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance Rev. Assoc. Française Finance 25, 71--94.

Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566--576.

Mathematical Reviews (MathSciNet): MR1149016

Digital Object Identifier: doi:10.1112/jlms/s2-44.3.566

Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.

Mathematical Reviews (MathSciNet): MR210154

Furrer, H. (1998). Risk processes perturbed by $\alpha$-stable Lévy motion. Scand. Actuarial J. 1, 59--74.

Mathematical Reviews (MathSciNet): MR1626676

Harrison, J. M. (1977). The supremum distribution of a Lévy process with no negative jumps. Adv. Appl. Prob. 9, 417--422.

Mathematical Reviews (MathSciNet): MR445619

Digital Object Identifier: doi:10.2307/1426392

JSTOR: links.jstor.org

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.

Mathematical Reviews (MathSciNet): MR2071427

Digital Object Identifier: doi:10.1214/105051604000000332

Project Euclid: euclid.aoap/1089736289

Zentralblatt MATH: 1061.60075

Kemperman, J. (1961). The Passage Problem for a Stationary Markov Chain. University of Chicago Press.

Mathematical Reviews (MathSciNet): MR119226

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.

Mathematical Reviews (MathSciNet): MR2099651

Digital Object Identifier: doi:10.1214/105051604000000927

Project Euclid: euclid.aoap/1099674077

Korevaar, J. (2002). A century of complex Tauberian theory. Bull. Amer. Math. Soc. (N.S.) 39, 475--531.

Mathematical Reviews (MathSciNet): MR1920279

Digital Object Identifier: doi:10.1090/S0273-0979-02-00951-5

Zentralblatt MATH: 1001.40007

Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2250061

Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 4, 473--493.

Mathematical Reviews (MathSciNet): MR1932381

Digital Object Identifier: doi:10.1007/s007800200070

Mordecki, E. (2002). The distribution of the maximum of a Lévy process with positive jumps of phase-type. Theory Stoch. Process. 8, 309--316.

Mathematical Reviews (MathSciNet): MR2027403

Mordecki, E. (2003). Ruin probabilities for Lévy processes with mixed-exponential negative jumps. Theory Prob. Appl. 48, 170--176.

Mathematical Reviews (MathSciNet): MR2013414

Petrov, V. (1987). Limit Theorems for Sums of Independent Random Variables. Nauka, Moscow (in Russian).

Mathematical Reviews (MathSciNet): MR896036

Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy process. J. Appl. Prob. 43, 208--220.

Mathematical Reviews (MathSciNet): MR2225061

Digital Object Identifier: doi:10.1239/jap/1143936254

Project Euclid: euclid.jap/1143936254

Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 21--34.

Mathematical Reviews (MathSciNet): MR740248

Rogozin, B. (1966). On distributions of functionals related to boundary problems for processes with independent increments. Theory Prob. Appl. 11, 580--591.

Mathematical Reviews (MathSciNet): MR208682

Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies Adv. Math. 68). Cambridge University Press.

Mathematical Reviews (MathSciNet): MR1739520

Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 65, 243--256.

Mathematical Reviews (MathSciNet): MR1875147

Digital Object Identifier: doi:10.1112/S0024610701002885

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.

Mathematical Reviews (MathSciNet): MR171315

previous :: next