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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
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Fluctuations of Lévy Processes with Applications
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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.
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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.
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Density and tails of unimodal convolution semigroups
TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
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.
References
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Journal ArticleDOI
On the exit time from open sets of some semi-markov processes
TL;DR: In this paper, the authors characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes.
Proceedings ArticleDOI
Relations between information and estimation in scalar Lévy channels
TL;DR: In this paper, the authors introduce the natural family of scalar Lévy channels where the distribution of the output conditioned on the input is infinitely divisible, and establish new representations relating the mutual information between the channel input and output to an optimal estimation loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings.
Journal ArticleDOI
Tail Probabilities of Subadditive Functionals of Lévy Processes
TL;DR: In this paper, the authors studied the tail behavior of the distribution of certain subadditive functionals acting on the sample paths of Levy processes, and showed that only the points of the process that lie above a certain curve contribute to the value of the functional.
Journal ArticleDOI
Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $\alpha$-stable L\'{e}vy processes
Robert Hintze,Ilya Pavlyukevich +1 more
TL;DR: In this article, the authors studied the asymptotic behavior of one-dimensional integrated Ornstein-Uhlenbeck processes in the Skorokhod $M_1$-topology.
Journal ArticleDOI
Extreme statistics of anomalous subdiffusion following a fractional Fokker-Planck equation: subdiffusion is faster than normal diffusion
TL;DR: In this article, the authors investigated extreme statistics of searchers which move by anomalous subdiffusion and proved that extreme FPTs of sub-diffusion are faster than normal FPT of normal diffusion.