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

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.

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

Acta Applicandae Mathematicae ◽

10.1007/s10440-019-00288-8 ◽

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.

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Applications of factorization embeddings for Lévy processes

Advances in Applied Probability ◽

10.1017/s0001867800001269 ◽

2006 ◽

Vol 38 (03) ◽

pp. 768-791 ◽

Cited By ~ 1

Author(s):

A. B. Dieker

Keyword(s):

Lévy Process ◽

Joint Distribution ◽

Lévy Processes ◽

Levy Processes ◽

Levy Process ◽

Tail Asymptotics ◽

Lévy Measure ◽

Risk Models ◽

General Properties ◽

Phase Type

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

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

Journal of Applied Probability ◽

10.1017/jpr.2019.62 ◽

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.

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Error Bounds for Small Jumps of Lévy Processes

Advances in Applied Probability ◽

10.1239/aap/1363354104 ◽

2013 ◽

Vol 45 (1) ◽

pp. 86-105

Author(s):

E. H. A. Dia

Keyword(s):

Monte Carlo ◽

Brownian Motion ◽

Error Bounds ◽

Monte Carlo Methods ◽

Lévy Process ◽

Lévy Processes ◽

Levy Processes ◽

Levy Process ◽

Lévy Measure ◽

Exponential Lévy Models

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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LÉVY PROCESSES IN ANIMAL MOVEMENT: AN EVOLUTIONARY HYPOTHESIS

Fractals ◽

10.1142/s0218348x07003460 ◽

2007 ◽

Vol 15 (02) ◽

pp. 151-162 ◽

Cited By ~ 98

Author(s):

FREDERIC BARTUMEUS

Keyword(s):

Lévy Processes ◽

Animal Movement ◽

Fundamental Problem ◽

Random Search ◽

Levy Processes ◽

Search Problem ◽

Animal Movements ◽

Adaptive Mechanisms ◽

Wide Range ◽

Fractal Patterns

The origin of fractal patterns is a fundamental problem in many areas of science. In ecological systems, fractal patterns show up in many subtle ways and have been interpreted as emergent phenomena related to some universal principles of complex systems. Recently, Lévy-type processes have been pointed out as relevant in large-scale animal movements. The existence of Lévy probability distributions in the behavior of relevant variables of movement, introduces new potential diffusive properties and optimization mechanisms in animal foraging processes. In particular, it has been shown that Lévy processes can optimize the success of random encounters in a wide range of search scenarios, representing robust solutions to the general search problem. These results set the scene for an evolutionary explanation for the widespread observed scale-invariant properties of animal movements. Here, it is suggested that scale-free reorientations of the movement could be the basis for a stochastic organization of the search whenever strongly reduced perceptual capacities come into play. Such a proposal represents two new evolutionary insights. First, adaptive mechanisms are explicitly proposed to work on the basis of stochastic laws. And second, though acting at the individual-level, these adaptive mechanisms could have straightforward effects at higher levels of ecosystem organization and dynamics (e.g. macroscopic diffusive properties of motion, population-level encounter rates). Thus, I suggest that for the case of animal movement, fractality may not be representing an emergent property but instead adaptive random search strategies. So far, in the context of animal movement, scale-invariance, intermittence, and chance have been studied in isolation but not synthesized into a coherent ecological and evolutionary framework. Further research is needed to track the possible evolutionary footprint of Lévy processes in animal movement.

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Cópulas Dinâmicas de Lévy e suas Aplicações no Apreçamento de Opções Multidimensionais com Dependência na Trajetória

Brazilian Review of Finance ◽

10.12660/rbfin.v6n1.2008.1234 ◽

2008 ◽

Vol 6 (1) ◽

pp. 69

Author(s):

Edson Bastos e Santos ◽

Nelson Ithiro Tanaka

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Asset Pricing ◽

Lévy Processes ◽

Levy Processes ◽

Financial Assets ◽

Lévy Measure ◽

Underlying Asset ◽

Price Process ◽

Gamma Processes

This article presents an alternative to modeling multidimensional options, where the payoffs depend on the paths of the trajectories of the underlying-asset prices. The proposed technique considers Lévy processes, a very ample class of stochastic processes that allows the existence of jumps (discontinuities) in the price process of financial assets, and as a particular case, comprises the Brownian motion. To describe the dependence among Lévy processes, extending the static concepts of the ordinary copulas to the Lévy processes context, considering the Lévy measure, which characterizes the jumps behavior of these processes. A comparison between the Clayton and the Frank dynamic copulas and their impact in asset pricing of Asian type derivatives contracts is studied, considering gamma processes and Monte Carlo simulation procedures.

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Applications of factorization embeddings for Lévy processes

Advances in Applied Probability ◽

10.1239/aap/1158685001 ◽

2006 ◽

Vol 38 (3) ◽

pp. 768-791 ◽

Cited By ~ 10

Author(s):

A. B. Dieker

Keyword(s):

Lévy Process ◽

Joint Distribution ◽

Lévy Processes ◽

Levy Processes ◽

Levy Process ◽

Tail Asymptotics ◽

Lévy Measure ◽

Risk Models ◽

General Properties ◽

Phase Type

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

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

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000062 ◽

2013 ◽

Vol 94 (3) ◽

pp. 304-320 ◽

Cited By ~ 2

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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More on hypergeometric Lévy processes

Advances in Applied Probability ◽

10.1017/apr.2016.47 ◽

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.

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