The Annals of Applied Probability

Meromorphic Lévy processes and their fluctuation identities

A. Kuznetsov, A. E. Kyprianou, and J. C. Pardo

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The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119–146, Electron. J. Probab. 13 (2008) 1672–1701], hyper-exponential and generalized hyper-exponential Lévy processes [Quant. Finance 10 (2010) 629–644], Lamperti-stable processes in [J. Appl. Probab. 43 (2006) 967–983, Probab. Math. Statist. 30 (2010) 1–28, Stochastic Process. Appl. 119 (2009) 980–1000, Bull. Sci. Math. 133 (2009) 355–382], Hypergeometric processes in [Ann. Appl. Probab. 20 (2010) 522–564, Ann. Appl. Probab. 21 (2011) 2171–2190, Bernoulli 17 (2011) 34–59], β-processes in [Ann. Appl. Probab. 20 (2010) 1801–1830] and θ-processes in [J. Appl. Probab. 47 (2010) 1023–1033].

In this paper we introduce a new family of Lévy processes, which we call Meromorphic Lévy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener–Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. The specific structure of the M-class Wiener–Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

Article information

Ann. Appl. Probab., Volume 22, Number 3 (2012), 1101-1135.

First available in Project Euclid: 18 May 2012

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Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G50: Sums of independent random variables; random walks

Lévy processes Wiener–Hopf factorization exit problems fluctuation theory


Kuznetsov, A.; Kyprianou, A. E.; Pardo, J. C. Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 (2012), no. 3, 1101--1135. doi:10.1214/11-AAP787.

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  • [1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15 2062–2080.
  • [2] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14 215–238.
  • [3] Baurdoux, E. J. (2009). Some excursion calculations for reflected Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 6 149–162.
  • [4] Baurdoux, E. J. and Kyprianou, A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Probab. Appl. 53 481–499.
  • [5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [6] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 156–169.
  • [7] Biffis, E. and Kyprianou, A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46 85–91.
  • [8] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Applications of Mathematics 4. Springer, New York.
  • [9] Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 967–983.
  • [10] Caballero, M. E., Pardo, J. C. and Perez, J. L. (2010). On the Lamperti stable processes. Probab. Math. Statist. 30 1–28.
  • [11] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13 345–382.
  • [12] Čebotarev, N. G. and Meĭman, N. N. (1949). The Routh–Hurwitz problem for polynomials and entire functions. Real quasipolynomials with r = 3, s = 1. Tr. Mat. Inst. Steklova 26 331.
  • [13] Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 980–1000.
  • [14] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin.
  • [15] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91–106.
  • [16] Es-Saghouani, A. and Mandjes, M. (2008). On the correlation structure of a Lévy-driven queue. J. Appl. Probab. 45 940–952.
  • [17] Gerber, H. U. and Shiu, E. S. W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math. Econom. 21 129–137.
  • [18] Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75–90.
  • [19] Hilberink, B. and Rogers, L. C. G. (2002). Optimal capital structure and endogenous default. Finance Stoch. 6 237–263.
  • [20] Hubalek, F. and Kyprianou, A. E. (2011). Old and new examples of scale functions for spectrally negative Lévy processes. In Sixth Seminar on Stochastic Analysis, Random Fields and Applications (R. Dalang, M. Dozzi and F. Russo, eds.) 119–146. Birkhäuser, Boston.
  • [21] Hurd, T. R. and Kuznetsov, A. (2008). Explicit formulas for Laplace transforms of stochastic integrals. Markov Process. Related Fields 14 277–290.
  • [22] Iserles, A. (2004). On the numerical quadrature of highly-oscillating integrals. I. Fourier transforms. IMA J. Numer. Anal. 24 365–391.
  • [23] 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.
  • [24] Kadankov, V. F. and Kadankova, T. V. (2005). On the distribution of the first exit time from an interval and the value of the overjump across a boundary for processes with independent increments and random walks. Ukraïn. Mat. Zh. 57 1359–1384.
  • [25] Kadankova, T. and Veraverbeke, N. (2007). On several two-boundary problems for a particular class of Lévy processes. J. Theoret. Probab. 20 1073–1085.
  • [26] Konstantopoulos, T., Kyprianou, A. E., Salminen, P. and Sirviö, M. (2008). Analysis of stochastic fluid queues driven by local-time processes. Adv. in Appl. Probab. 40 1072–1103.
  • [27] Kou, S. (2002). A jump diffusion model for option pricing. Management Science 48 1086–1101.
  • [28] Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 1801–1830.
  • [29] Kuznetsov, A. (2010). Wiener–Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Probab. 47 1023–1033.
  • [30] 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. Probab. 21 2171–2190.
  • [31] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [32] Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab. 20 522–564.
  • [33] Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 1672–1701.
  • [34] Kyprianou, A. E. and Surya, B. A. (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11 131–152.
  • [35] Levin, B. Y. (1996). Lectures on Entire Functions. Translations of Mathematical Monographs 150. Amer. Math. Soc., Providence, RI.
  • [36] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Probab. 45 118–134.
  • [37] McKean, H. (1965). Appendix: A free boundary problem for the heat equation arising from a problem of mathematical economics. Ind. Manag. Rev. 6 32–39.
  • [38] Rogers, L. C. G. (1983). Wiener–Hopf factorization of diffusions and Lévy processes. Proc. Lond. Math. Soc. (3) 47 177–191.
  • [39] Rogozin, B. A. (1972). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theor. Probab. Appl. 17 332–338.
  • [40] Schilling, R. L., Song, R. and Vondraček, Z. (2010). Bernstein Functions: Theory and Applications. de Gruyter Studies in Mathematics 37. de Gruyter, Berlin.
  • [41] Vigon, V. (2002). Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf. Thèse de doctorat de l’INSA de Rouen.
  • [42] Wang, S. and Zhang, C. (2007). First-exit times and barrier strategy of a diffusion process with two-sided jumps. Preprint.