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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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The time at which a L\'evy process creeps
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TL;DR: In this paper, it was shown that the renewal function of the bivariate ascending ladder process (L^{-1},H) is continuous on $[0,\infty) and left differentiable on $ [0, \infty] and that the left derivative at $u$ is proportional to the (improper) distribution function of time at which the process creeps over level $u$, where the constant of proportionality is the reciprocal of the (positive) drift of $H$.
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On the infinite divisibility of inverse Beta distributions
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TL;DR: In this paper, it was shown that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering a recent question of L. Bondesson, who showed that negative powers are not always a generalized gamma convolution.
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The analog formulation of sparsity implies infinite divisibility and rules out Bernoulli-Gaussian priors
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Fractional Fokker-Planck-Kolmogorov equations associated with SDES on a bounded domain
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The exit problem from the neighborhood of a global attractor for heavy-tailed Lévy diffusions
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TL;DR: In this article, the authors considered a deterministic dynamical system with a global attractor A and a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A.
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.