We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in [25] and [23]. We also derive some new results related to (i) the entrance law of the stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of the stable process reflected at its past infimum and (iii) the entrance law and the last passage time of the radial part of n-dimensional symmetric stable process.

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 [21,32], hyper-exponential and generalized hyper-exponential Lévy processes [24], Lamperti-stable processes in [9,10,13,39], Hypergeometric processes in [35,31,11], β-processes in [29] and θ-processes in [30].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.

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr's so-called "Canadization" technique as well as Doney's method of stochastic bounds for Lévy processes; see Carr [Rev. Fin. Studies 11 (1998) 597-626] and Doney [Ann. Probab. 32 (2004) 1545-1552]. We rely fundamentally on the Wiener-Hopf decomposition for Lévy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Lévy en titillant la factorization de Wiener-Hopf (2002) Laboratoire de Mathématiques de L'INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801-1830]. We illustrate our Wiener-Hopf Monte Carlo method on a number of different processes, including a new family of Lévy processes called hypergeometric Lévy processes. Moreover, we illustrate the robustness of working with a Wiener-Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Lévy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type π(dx) = e γx ν(e x − 1) dx, where ν is the density of the stable Lévy measure and γ is a positive parameter which depends on its characteristics. These processes were introduced in [10] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.