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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
Citations
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Journal ArticleDOI
Densities of scaling limits of coupled continuous time random walks
Marcin Magdziarz,Tomasz Zorawik +1 more
TL;DR: In this paper, the densities of Lévy walks are derived for both jump-first and wait-first scenarios, where the stability index is rational and densities can be represented as an integral of Meijer G-function.
Journal ArticleDOI
Burkholder-Davis-Gundy inequalities in UMD Banach spaces
TL;DR: In this article, it was shown that for general martingales with values in a UMD Banach space, the right-hand side of \eqref{eq:main} can be expressed more explicitly in terms of the jumps of the martingale.
Journal ArticleDOI
Local time of Lévy random walks: A path integral approach
TL;DR: This work focuses on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations and uses the phase-space path-integral representation of random walk transition probabilities to quantify the properties of the local time.
Posted Content
Rogers functions and fluctuation theory
TL;DR: In this article, the Wiener-Hopf factorisation for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions is studied, and a semi-explicit expression for the space-only Laplace transform of the supremum and the infimum of X_t follows.
Posted Content
The Small Maturity Implied Volatility Slope for L\'evy Models
TL;DR: In this article, the authors considered the at-the-money strike derivative of implied volatility as the maturity tends to zero and quantified the growth of the slope for infinite activity exponential Levy models.
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.