Meromorphic Lévy processes and their fluctuation identities

Abstract

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.

The Theory of Scale Functions for Spectrally Negative Lévy Processes

“…See for example Corollary 2 and Remark 2 in [65] as well as [66]. The second statement can be established using the same technique as in the proof of Theorem 1 in [64].…”

“…On the negative side, it is clear that these processes can only have compound Poisson jumps, while in applications it is often necessary to have processes with jumps of infinite activity or even of infinite variation. Meromorphic Lévy processes, which were recently introduced in [65], solve precisely this problem. They allow for much more flexible modeling of the small jump behavior, yet all the computations can be done with the same efficiency as for processes with jumps of rational transform.…”

Pricing of contingent convertibles under smile conform models

“…For the case of β-VG processes, one can develop the decomposition (1) further by specifying the distribution of the variables X e(q) and X e(q) explicitly; cf. Kuznetsov et al [14]. Indeed, it can be shown (cf.…”

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

“…The claim of the statement follows by noting that (20) and (21) are upper bounds independent from τ u and applying the tower property in (19).…”

“…This is a subclass, like the θ -processes (see [19]) and general hypergeometric Lévy processes (see [21]), of the family of so-called meromorphic Lévy processes as recently introduced in [20]. The class of meromorphic Lévy processes is very rich: paths of bounded and unbounded variation and both finite and infinite activity jumps can be generated.…”