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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
Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results
TL;DR: Meerschaert et al. as discussed by the authors studied the fractional Cauchy problem on bounded domains, where the first time derivative is replaced with an infinite sum of fractional derivatives.
Journal ArticleDOI
Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems
Carles Bretó,Edward L. Ionides +1 more
TL;DR: In this article, the authors proposed an infinitesimal dispersion index for Markov counting processes, which increases in jumps of one (or more) unit(s), even though the processes might be under-, equi- or over-dispersed using previously studied indices.
Journal ArticleDOI
Reflected Spectrally Negative Stable Processes and their Governing Equations
TL;DR: In this article, the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum, were derived using the theory of sun-dual semigroups.
Journal ArticleDOI
Continuous dependence estimates for nonlinear fractional convection-diffusion equations
TL;DR: Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary.
Posted Content
A guide to Brownian motion and related stochastic processes
Jim Pitman,Marc Yor +1 more
TL;DR: The mathematical theory of Brownian motion and related stochastic processes is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof as mentioned in this paper.
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.