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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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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.
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.
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Density and tails of unimodal convolution semigroups
TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
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.
References
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Kato Classes for Lévy Processes
TL;DR: In this article, it was shown that the definitions of the Kato class through the semigroup and through the resolvent of the Levy process in $\mathbb {R}^{d}$ coincide if and only if 0 is not regular for {0}.
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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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Distance multivariance: New dependence measures for random vectors
TL;DR: In this paper, distance covariance and total distance multivariance have been introduced to detect the independence of random variables and have a simple finite sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function.
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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.