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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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Integral criteria for Strong Renewal Theorems with infinite mean
TL;DR: In this paper, a probabilistic approach to Strong Renewal Theorem (SRT) was proposed, and integral criteria for the SRT were established for the ladder height process of a random walk with step distribution.
Journal ArticleDOI
Convex semigroups on $$L^p$$ L p -like spaces
TL;DR: In this paper, it was shown that the generator of a convex convex semigroup is closed and uniquely determines the semigroup whenever the domain is dense, and that the domain of the generator is invariant under the semi-giver.
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Scattering length for stable processes
TL;DR: In this article, the first eigenvalue of the Schr\"odinger operator of the ''Neumann'' fractional Laplacian in a cube with potential $v$ was estimated.
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Contemporaneous aggregation of triangular array of random-coefficient AR(1) processes
TL;DR: In this paper, the authors discuss contemporaneous aggregation of independent copies of a triangular array of random-coefficient AR(1) processes with i.i.d. innovations belonging to the domain of attraction of an infinitely divisible law W. The authors show that the above partial sums process may exhibit four different limit behaviors depending on \beta and the L\'evy triplet of W.
Proceedings ArticleDOI
Stochastic differential equations for power law behaviors
TL;DR: A model with bi-directional Poisson counters that exhibits power law behavior near a critical point, which might be of interest to statistical physics.
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.