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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
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Positive stable densities and the bell-shape
TL;DR: In this paper, it was shown that positive stable densities are bell-shaped, that is their n-th derivatives vanish exactly n times on (0,+oo) and have an alternating sign sequence.
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On the range of exponential functionals of L\'evy processes
TL;DR: In this paper, the authors characterized the support of the law of the exponential functional of two one-dimensional independent L\'evy processes and showed that the range of this mapping is closed under weak convergence.
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Subexponential loss rate asymptotics for Lévy processes
TL;DR: This work derives asymptotics for the loss rate when K tends to infinity, when the mean of the Lévy process is negative and the positive jumps are subexponential, and achieves a formula which is a generalization of the celebrated Pollaczeck-Khinchine formula.
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Limit theorems for the sample mean and sample autocovariances of continuous time moving averages driven by heavy-tailed Levy noise
TL;DR: In this article, the authors consider continuous time moving averages observed on a lattice, driven by an infinite variance Lévy process with regularly varying tails with index α ∈ (0, 2).
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Unimodality of Boolean and monotone stable distributions
Takahiro Hasebe,Noriyoshi Sakuma +1 more
TL;DR: A complete list of the Lebesgue-Jordan decomposition of stable distributions of Boolean and monotone stable distributions is given in this article, along with a list of their modes.
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.