The distribution of the supremum for spectrally asymmetric L'evy processes (original) (raw)

A Distributional Equality for Suprema of Spectrally Positive Lévy Processes

Journal of Theoretical Probability, 2015

Let Y be a spectrally positive Lévy process with EY 1 < 0, C an independent subordinator with finite expectation, and X = Y + C. A curious distributional equality proved in [3] states that if EX 1 < 0, then sup 0≤t<∞ Y t and the supremum of X just before the first time its new supremum is reached by a jump of C have the same distribution. In this paper we give an alternative proof of an extension of this result and offer an explanation why it is true.

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Lecture Notes in Mathematics, 2012

The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes;

On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum

Journal of Theoretical Probability, 2004

Consider a spectrally one-sided Levy process X and reflect it at its past infimum I. Call this process Y . For spectrally positive X, Avram et al. [2] found an explicit expression for the law of the first time that Y = X I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y , if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0, a]. We determine the exponential decay parameter # for the transition probabilities of Y killed upon leaving [0, a], prove that this killed process is #-positive and specify the #-invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined to [0, a] and proof some properties of this process.

On the density of exponential functionals of Lévy processes

Bernoulli, 2013

In this paper, we study the existence of the density associated to the exponential functional of the Lévy process ξ, I eq := eq 0 e ξs ds, where e q is an independent exponential r.v. with parameter q ≥ 0. In the case when ξ is the negative of a subordinator, we prove that the density of I eq , here denoted by k, satisfies an integral equation that generalizes the one found by Carmona et al. [7]. Finally when q = 0, we describe explicitly the asymptotic behaviour at 0 of the density k when ξ is the negative of a subordinator and at ∞ when ξ is a spectrally positive Lévy process that drifts to +∞.

Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2012

We determine the rate of decrease of the right tail distribution of the exponential functional of a Lévy process with a convolution equivalent Lévy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Lévy measure of the underlying Lévy process. The method of proof relies on fluctuation theory of Lévy processes and an explicit pathwise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish analogous results under the excursion measure of the underlying Lévy process reflected in its past infimum. Résumé. On s'intéresse à la vitesse de décroissance de la queue de distribution d'une fonctionnelle exponentielle d'un processus de Lévy dont la mesure de sauts est équivalente par convolution. Le résultat principal de ce papier montre que cette vitesse décroît comme la queue de la mesure image de la mesure de sauts par la fonction exponentielle. La preuve de ce résultat repose sur la théorie des fluctuations pour les processus de Lévy et une représentation trajectorielle explicite de la fonctionnelle exponentielle comme la fonctionnelle exponentielle d'un subordinateur bivarié. Nos techniques nous permettent également d'établir des résultats similaires sous la mesure d'excursion du processus de Lévy sous-jacent réfléchi en son minimum passé.

On suprema of Lévy processes and application in risk theory

Annales de l'Institut Henri Poincaré, …, 2008

Let X = C − Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X is reached by a jump of the subordinator C. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and X drifts to −∞, we decompose the absolute supremum of X at these times, and derive a Pollaczek-Hinchintype formula for the distribution function of the supremum. Soit Y un processus de Lévy réel quelconque et C un subordinateur indépendant de Y. On considère les temps en lesquels le processus X = C − Y atteint un nouveau maximum par un saut de C. Nous donnons une condition nécessaire et suffisante pour que l'ensemble de ces temps soit discret. Lorsque tel est le cas et que le processus X dérive vers −∞, nous décomposons son maximum absolu en cette suite de temps. Nous déduisons alors de cette décomposition une formule du type Pollaczek-Hinchin pour la loi du maximum absolu de X.

Smoothness of scale functions for spectrally negative Lévy processes

Probability Theory and Related Fields, 2011

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation.

Fluctuations of Omega-killed spectrally negative Lévy processes

Stochastic Processes and their Applications

In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases ω(x) = q and ω(x) = q1 (a,b) (x), we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate ω(x) when the Lévy surplus process is at level x < 0. Finally, we apply the these results to obtain some exit identities for a spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.

On Scale Functions for Spectrally Negative Lévy Processes with Phase-type Jumps

We study the scale function for the class of spectrally negative Lévy processes with phasetype jumps. We consider both the compound Poisson case and the unbounded variation case with diffusion components, and obtain the corresponding scale functions explicitly. Motivated by the fact that the class of phase-type distributions is dense in the class of all positive-valued distributions, we propose a new approach to approximating the scale function for a general spectrally negative Lévy process. We illustrate, in numerical examples, its effectiveness by obtaining the scale functions for Lévy processes with long-tail distributed jumps.

Distributional properties of exponential functionals of Lévy processes

Electronic Journal of Probability, 2012

We study the distribution of the exponential functional I(ξ, η) = ∞ 0 exp(ξ t− )dη t , where ξ and η are independent Lévy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in . In the special case when η is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of I(ξ, η), which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy process ξ (η being Brownian motion with drift) we prove that the exponential functional has a smooth density on R\{0}, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that ξ has some positive exponential moments we establish an asymptotic behaviour of P(I(ξ, η) > x) as x → +∞, and under similar assumptions on the negative exponential moments of ξ we obtain a precise asympotic expansion of the density of I(ξ, η) as x → 0. Under further assumptions on the Lévy process ξ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when ξ has hyper-exponential jumps.