Juan Carlos Pardo

Meromorphic Lévy 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], 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 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 theirWiener.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.

Exact and asymptotic n-tuple laws at first and last passage

Understanding the space time features of how a Levy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91-106] a new quintuple law was established for a general Levy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Levy processes, Levy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Levy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Levy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Levy processes. This leads further to the introduction of a more general family of Levy processes which we call hypergeometric Levy processes, for which similar explicit identities may be considered.

Fluctuation theory and exit systems for positive self-similar Markov processes

For a positive self-similar Markov process, X, we construct a local time for the random set, Θ, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set Θ and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Levy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Levy process oscillates.

Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

Inspired by works of Landriault et al. [11, 12], we study the Gerber{Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Levy insurance risk process. To be more specific, we study the so-called Gerber{Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Levy processes and relies on the theory of so-called scale functions. In particular, we extend recent results of Landriault et al. [11, 12].

The hitting time of zero for a stable process

For any two-sided jumping

HITTING DISTRIBUTIONS OF -STABLE PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY

\alpha

On the density of exponential functionals of Levy processes

-stable process, where

A Wiener-Hopf type factorization for the exponential functional of Lévy processes

1 < \alpha<2

Branching Processes in a Lévy Random Environment

, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Panti-Rivero (2011) for real-valued self similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.

The extended hypergeometric class of Lévy processes

After Brownian motion, �-stable processes are often considered an exemplary family of processes for which many aspects of the general theory of Lproesses can be illus- trated in closed form. First passage problems, which are relatively straightforward to handle in the case of Brownian motion, become much harder in the setting of a general Levy process on account of the inclusion of jumps. A collection of articles through the 1960s and early 1970s, appealing largely to potential analytic methods for general Markov processes, were relatively successful in handling a number of first passage prob- lems, in particular for symmetric �-stable processes in one or more dimensions. See for example (3, 24, 12, 13, 27) to name but a few. However, following this cluster of activity, several decades have passed since new results concerning first passage identities for �-stable processes have appeared. The last few years have seen a number of new, explicit first passage identities for one- dimensional �-stable processes thanks to a better understanding of the intimate re- lationship between the aforesaid processes and positive self-similar Markov processes. See for example (4, 6, 8, 17, 22). In this paper we return to the problem of Blumenthal et al. (3), published in 1961, which gave the law of the position of first entry of a symmetric �-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional �-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between �-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for �-stable processes.