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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
Extremal theory for long range dependent infinitely divisible processes
Gennady Samorodnitsky,Yizao Wang +1 more
TL;DR: In this article, the authors prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes, which involve multiple phase transitions governed by how long the memory is.
Journal ArticleDOI
Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
TL;DR: In this paper, it was shown that the implied volatility has a uniform (in log moneyness x) limit as the maturity tends to infinity, given by an explicit closed-form formula, for x in some compact neighborhood of zero in the class of affine stochastic volatility models.
Journal ArticleDOI
Generalized fractional Lévy processes with fractional Brownian motion limit
TL;DR: In this article, the authors extend the usual fractional Riemann-Liouville kernel to a regularly varying function and show that the resulting stochastic processes can have short or long memory increments and their sample paths may have jumps or not.
Journal ArticleDOI
Semi-Markov Models and Motion in Heterogeneous Media
Costantino Ricciuti,Bruno Toaldo +1 more
TL;DR: In this article, the authors studied continuous time random walks such that the holding time in each state has a distribution depending on the state itself, and provided integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel.
Journal ArticleDOI
A law of large numbers for the power variation of fractional Lévy processes
TL;DR: In this article, a law of large numbers for the power variation of an integrated fractional process in a pure jump model has been proved, which yields consistency of an estimator for the integrated volatility where we are no longer restricted to a Gaussian model.
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.