scholarly journals The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

2019 ◽  
Vol 169 (1) ◽  
pp. 59-77
Author(s):  
Loïc Chaumont ◽  
Jacek Małecki
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Time Variable ◽  
Stable Processes ◽  
The Laplace Transform ◽  
Entrance Law ◽  
Generalized Eigenfunctions ◽  
Positive Half ◽  
Integral Formulae

Abstract We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

scholarly journals More on hypergeometric Lévy processes

2016 ◽  
Vol 48 (A) ◽  
pp. 153-158
Author(s):  
Emma L. Horton ◽  
Andreas E. Kyprianou
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Stable Processes ◽  
Short Article ◽  
The Family ◽  
Self Similar ◽  
Definition Of ◽  
Complex Valued ◽  
Independent Parameters

AbstractKuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

scholarly journals Oscillation of modes of some semi-stable Lévy processes

1993 ◽  
Vol 132 ◽  
pp. 141-153 ◽  
Author(s):  
Toshiro Watanabe
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Processes ◽  
Unimodal Lévy Process ◽  
Oscillating Mode

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (04) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

scholarly journals The distribution and asympotic behaviour of the negative Wiener–Hopf factor for Lévy processes with rational positive jumps

2019 ◽  
Vol 56 (4) ◽  
pp. 1086-1105
Author(s):  
Ekaterina T. Kolkovska ◽  
Ehyter M. Martín-González
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Lévy Processes ◽  
Levy Processes ◽  
Regularity Conditions ◽  
Asymptotic Results ◽  
Lévy Measure ◽  
The Laplace Transform ◽  
Additional Regularity

AbstractWe study the distribution of the negative Wiener–Hopf factor for a class of two-sided jump Lévy processes whose positive jumps have a rational Laplace transform. The positive Wiener–Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener–Hopf factor, as well as an explicit expression for its probability density in terms of sums of convolutions of known functions. Under additional regularity conditions on the Lévy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function F(u) of the negative Wiener–Hopf factor. We illustrate our results in some particular examples.

scholarly journals Conditioned stable Lévy processes and the Lamperti representation

2006 ◽  
Vol 43 (04) ◽  
pp. 967-983 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Lévy Process ◽  
Self Similar ◽  
Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

scholarly journals Double hypergeometric Lévy processes and self-similarity

2021 ◽  
Vol 58 (1) ◽  
pp. 254-273
Author(s):  
Andreas E. Kyprianou ◽  
Juan Carlos Pardo ◽  
Matija Vidmar
Keyword(s):  
Closed Form ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stable Processes ◽  
Self Similarity ◽  
New Family ◽  
Self Similar

AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

scholarly journals Conditioned stable Lévy processes and the Lamperti representation

2006 ◽  
Vol 43 (4) ◽  
pp. 967-983 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Lévy Process ◽  
Self Similar ◽  
Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

scholarly journals ASPECTS OF RECURRENCE AND TRANSIENCE FOR LÉVY PROCESSES IN TRANSFORMATION GROUPS AND NONCOMPACT RIEMANNIAN SYMMETRIC PAIRS

2013 ◽  
Vol 94 (3) ◽  
pp. 304-320 ◽  
Author(s):  
DAVID APPLEBAUM
Keyword(s):  
Riemannian Manifolds ◽  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transformation Groups ◽  
Compact Sets ◽  
Topological Transformation ◽  
Local Integrability ◽  
Potential Measure ◽  
Complete Riemannian Manifolds

AbstractWe study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

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