MEROMORPHIC LEVY PROCESSES AND THEIR FLUCTUATION IDENTITIES

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Levy 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 Levy 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], beta-processes in [Ann. Appl. Probab. 20 (2010) 1801-1830] and theta-processes in [J. Appl. Probab. 47 (2010) 1023-1033]. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy 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.