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Lévy processes and infinitely divisible distributions
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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
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L{\'e}vy processes: concentration function and heat kernel bounds
TL;DR: In this article, the authors investigate densities of vaguely continuous convolutional semigroups of probability measures and show that typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable.
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Detecting independence of random vectors I. Generalized distance covariance and Gaussian covariance
TL;DR: In this article, the authors show that distance covariance can be embedded into a more general framework based on symmetric Levy measures and the corresponding real-valued continuous negative definite functions.
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Aggregation and long memory: recent developments
TL;DR: Recent work on contemporaneous aggregation of random-coefficient AR(1) and related models is reviewed, with particular focus on various long memory properties of the aggregated process.
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A Donsker Theorem for L\'evy Measures
Richard Nickl,Markus Reiß +1 more
TL;DR: In this article, a functional central limit theorem for a generalised Brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the Fourier-integral operator was proved.
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Thorin classes of Lévy processes and their transforms
TL;DR: In this paper, the authors define and characterize Thorin classes of infinitely divisible distributions on R+ and investigate Poisson, Karlin, and Bessel transforms of Thorin class.
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.