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Meromorphic Lévy processes and their fluctuation identities. (English) Zbl 1252.60044
The last couple of years have 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 [F. Hubalek and E. Kyprianou, in: R. C. Dalang (ed.) et al., Seminar on stochastic analysis, random fields and applications VI. Progress in Probability 63, 119–145 (2011; Zbl 1274.60148); A. E. Kyprianou and V. Rivero, Electron. J. Probab. 13, 1672–1701 (2008; Zbl 1193.60064)], hyper-exponential and generalized hyper-exponential Lévy processes [M. Jeannin and M. Pistorius, Quant. Finance 10, No. 6, 629–644 (2010; Zbl 1192.91177)], Lamperti-stable processes [M. E. Caballero and L. Chaumont, J. Appl. Probab. 43, No. 4, 967–983 (2006; Zbl 1133.60316); M. E. Caballero, J. C. Pardo and J. L. Pérez, Probab. Math. Stat. 30, No. 1, 1–28 (2010; Zbl 1198.60022); L. Chaumont, A. E. Kyprianou and J. C. Pardo, Stochastic Processes Appl. 119, No. 3, 980–1000 (2009; Zbl 1170.60017); P. Patie, Bull. Sci. Math. 133, No. 4, 355–382 (2009; Zbl 1171.60009)], Hypergeometric processes [A. E. Kyprianou, J. C. Pardo and V. Rivero, Ann. Appl. Probab. 20, No. 2, 522–564 (2010; Zbl 1200.60038); A. Kuznetsov, A. E. Kyprianou, J. C. Pardo and K. van Schaik, Ann. Appl. Probab. 21, No. 6, 2171–2190 (2011; Zbl 1245.65005); M. E. Caballero, J. C. Pardo and J. L. Pérez, Bernoulli 17, No. 1, 34–59 (2011; Zbl 1284.60092)], \(\beta\)-processes [A. Kuznetsov, Ann. Appl. Probab. 20, No. 5, 1801–1830 (2010; Zbl 1222.60038)] and \(\theta\)-processes [A. Kuznetsov, J. Appl. Probab. 47, No. 4, 1023–1033 (2010; Zbl 1223.60029)].
In their paper, the authors introduce a new family of Lévy processes, which they 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 the reader 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.

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