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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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Book ChapterDOI
Potential theory of subordinate Brownian motions revisited
TL;DR: In this paper, the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity is discussed.
Journal ArticleDOI
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations
TL;DR: In this article, it was shown that under a certain condition on a semimartingale and a time change, any stochastic integral driven by the time-changed semimARTingale is a time-changing SDE, and a specialized form of the Ito formula was derived.
Journal ArticleDOI
Stationarity and geometric ergodicity of BEKK multivariate GARCH models
TL;DR: In this paper, conditions for the existence of strictly stationary GARCH processes in the so-called BEKK parametrisation, which is the most general form of multivariate GARCH process typically used in applications, and for their geometric ergodicity are obtained.
Journal ArticleDOI
Multivariate supOU processes
TL;DR: In this article, the authors define multivariate supOU processes via homogeneous and factorizable Levy bases, and give conditions for their existence and finiteness of heartbeat moments, showing that the second-order moment structure is explicitly calculated.
Journal ArticleDOI
Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes
TL;DR: In this paper, the authors introduce the class of volatility modulated Levy-driven Volterra (VMLV) processes and their important subclass of Levy semistationary (LSS) processes as a new framework for modelling energy spot prices.
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.