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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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Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes
Eckhard Schlemm,Robert Stelzer +1 more
TL;DR: In this article, the class of multivariate Levy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the classes of continuous time state space models, and the linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular.
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Gradient estimates of harmonic functions and transition densities for Levy processes
Tadeusz Kulczycki,Michał Ryznar +1 more
TL;DR: In this article, the authors prove gradient estimates for harmonic functions with respect to a $d$-dimensional unimodal pure-jump Levy process under some mild assumptions on the density of its Levy measure.
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Asymptotic and Exact Pricing of Options on Variance
TL;DR: The small-time limits of options on both objects are characterized and it is found that the difference between them strongly depends on whether or not the stock price process has jumps.
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On finite truncation of infinite shot noise series representation of tempered stable laws
Junichi Imai,Reiichiro Kawai +1 more
TL;DR: In this paper, the authors derived series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Levy measure methods, and proved that the representation via the inverse Levy measures achieves a much faster convergence in truncation to the infinite sum than all the other representations.
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Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach
TL;DR: In this article, the authors proposed an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps.
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.