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Lévy processes and infinitely divisible distributions
TLDR
In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.Abstract:
Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.read more
Citations
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Journal ArticleDOI
The tempered stable process with infinitely divisible inverse subordinators
TL;DR: In this article, the authors extended the classical approach and replaced the subordinated Brownian motion by the tempered stable process and analyzed the general class of inverse subordinators, and examined the main properties of the tempered-stable process driven by inverse subordinator from the infinitely divisible class of distributions.
Journal ArticleDOI
Hitting times of points and intervals for symmetric L\'{e}vy processes
Tomasz Grzywny,Michał Ryznar +1 more
TL;DR: For one-dimensional symmetric L * evy processes, which hit every point with positive probability, this article gave sharp bounds for the tail function of the first hitting time of B which is either a single point or an interval, under some weak type scaling assumptions on the characteristic exponent of the process.
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On overload in a storage model, with a self-similar and infinitely divisible input
TL;DR: In this paper, the storage process of a locally bounded and infinitely divisible stochastic process, with no Gaussian component, was studied and the behavior of the following probability of overload during the time period was investigated.
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Convolution equivalent Lévy processes and first passage times
TL;DR: In this paper, the authors investigated the behavior of Levy processes with convolution equivalent Levy measures, up to the time of first passage over a high level $u$ such problems arise naturally in the context of insurance risk.
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Prediction of Lévy-driven CARMA processes
TL;DR: In this article, the conditional expectations for a continuous-time CARMA (CARMA) process (Y ( t ) ) t ∈ R driven by a Levy process L with E | L ( 1 ) | ∞.
References
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BookDOI
Fluctuations of Lévy Processes with Applications
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
Journal ArticleDOI
Ten equivalent definitions of the fractional laplace operator
TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Book ChapterDOI
The Theory of Scale Functions for Spectrally Negative Lévy Processes
TL;DR: In this article, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative Levy processes, including the first extensive overview of how to work numerically with scale functions.
Journal ArticleDOI
Optimal stopping and perpetual options for Lévy processes
TL;DR: A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formulas for perpetual American put options involving the infimum of the after-mentioned process are obtained.
Extreme Events: Dynamics, Statistics and Prediction
TL;DR: In this paper, the authors review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics.