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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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A distributional limit theorem for the realized power variation of linear fractional stable motions
TL;DR: In this article, the authors deduce a distributional theorem for the realized power variation of linear fractional stable motions by choosing the technique of subordination to reduce the proof to a Gaussian limit theorem based on Malliavin-calculus.
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Higher order Cauchy problems in bounded domains
TL;DR: In this article, Meerschaert, Nane and Vellaisamy study solutions of a class of higher-order partial differential equations in bounded domains, where the authors express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2.
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Ergodicities and exponential ergodicities of Dawson-Watanabe type processes
TL;DR: In this paper, the ergodicities of the Dawson-Watanabe superprocesses without or with immigration were derived under natural assumptions and the strong Feller property in the total variation distance was derived as a byproduct.
Journal Article
Characterizations of Multivariate Normal-Poisson Model
TL;DR: In this article, a multivariate normal-Poisson model was introduced as a special case of normal stable Tweedie models, which is composed of a univariate Poisson variable, and the remaining variables given the Poisson one are independent Gaussian variable with variance the value of the poisson component.
Journal ArticleDOI
Fractional generalizations of filtering problems and their associated fractional Zakai equations
TL;DR: In this article, the authors discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Levy process.
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.