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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
Efficient estimation of stable Lévy process with symmetric jumps
Alexandre Brouste,Hiroki Masuda +1 more
TL;DR: In this article, the asymptotic normality of the likelihood of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is investigated.
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Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes
TL;DR: In this paper, the existence of the weak limit of (z (x ), Z(x ) ) as x tends to infinity was established under the Cramer and positive drift assumptions.
Journal ArticleDOI
Multivariate stochastic integrals with respect to independently scattered random measures on $\delta$-rings
TL;DR: In this article, a general vector-valued infinite-divisible independently scattered random measure with values in $mathbb{R}^m$ and their corresponding stochastic integrals is constructed and a general construction principle is presented.
Journal ArticleDOI
Discretization error for a two-sided reflected Lévy process
Søren Asmussen,Jevgenijs Ivanovs +1 more
TL;DR: This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators L, U at the lower and upper barriers at some fixed time.
Journal ArticleDOI
A fractional Fokker–Planck control framework for subdiffusion processes
TL;DR: In this paper, an efficient framework for the optimal control of the probability density function of a sub-diffusion process is presented, which is based on a fractional Fokker-Planck equation that governs the time evolution of the PDF of the sub diffusion process and on tracking objectives of terminal configuration of the desired PDF.
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.