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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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Journal Article
Traveling waves for a bistable equation with nonlocal-diffusion
Franz Achleitner,Christian Kuehn +1 more
TL;DR: In this article, the authors considered a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type.
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Limit Theorems for Power Variations of Pure-Jump Processes with Application to Activity Estimation
Viktor Todorov,George Tauchen +1 more
TL;DR: In this article, the authors derived the asymptotic behavior of realized power variation of pure-jump Ito semimartingales as the sampling frequency within a fixed interval increases to infinity.
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Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration
TL;DR: In this paper, a least square estimator for the model parameter in a multivariate Ornstein-Uhlenbeck model driven by a multiivariate regularly varying Levy process with infinite variance is presented.
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Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise
TL;DR: In this article, the authors studied weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Levy noise and showed that the weak rate of convergence is twice the strong rate.
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Jump activity estimation for pure-jump semimartingales via self-normalized statistics
TL;DR: In this paper, the authors derived a nonparametric estimator of the jump activity index of a pure-jump It\€ o} semimartingale from discrete observations on a fixed time interval with mesh of the observation grid shrinking to zero.
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.