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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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Journal ArticleDOI
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.
Journal ArticleDOI
Realized Laplace transforms for pure-jump semimartingales
Viktor Todorov,George Tauchen +1 more
TL;DR: In this article, the authors consider the stochastic scale of discretely-observed pure-jump martingales with locally stable Levy densities in the setting where both the time span of the data set increases, and the mesh of the observation grid decreases.
Journal ArticleDOI
Monotone Increment Processes, Classical Markov Processes, and Loewner Chains
TL;DR: In this article, it was shown that decreasing Loewner chains in the upper half-plane correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.
Posted Content
Ornstein-Uhlenbeck processes driven by cylindrical L\'evy processes
TL;DR: In this article, a theory of integration for deterministic, operator-valued integrands with respect to cylindrical L\'evy processes in separable Banach spaces is introduced.
Journal ArticleDOI
Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
Thomas M. Michelitsch,Bernard Collet,Alejandro P. Riascos,Andrzej F. Nowakowski,Franck C. G. A. Nicolleau +4 more
TL;DR: In this paper, a fractional random walk strategy on undirected regular networks involving power matrix functions of the type $L^{\\frac{\\alpha}{2}}$ where $L$ indicates a ''simple'' Laplacian matrix.
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.