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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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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.
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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.
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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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Basics of Levy processes
TL;DR: In this paper, the authors present a book on Lévy driven volatility models (Lévy Driven Volatility Models) and present a draft chapter from a book by the authors.
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The dickman subordinator, renewal theorems, and disordered systems
TL;DR: In this article, the so-called Dickman subordinator, whose Levy measure has density δ(1/x) is restricted to the interval (0, 1) where x is the number of vertices in the Dickman subspace.
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On the density of the supremum of a stable process
TL;DR: In this article, the convergence properties of the series representation for the supremum of the strictly stable Levy process were investigated. But the convergence was not considered for the case when α is rational.
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Non-symmetric stable operators: regularity theory and integration by parts
TL;DR: In this article, the authors studied the regularity of solutions to the non-symmetric, stable Levy process in half-space and showed that the solutions satisfy the Holder regularity ε/d^\gamma in C^{\varepsilon_\circ}\big(\overline\Omega\big) where ε is the distance to the boundary.
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Fractional normal inverse Gaussian diffusion
TL;DR: In this article, the authors show that the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times, and that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit for random walks with uncorrelated jumps.