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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
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Moments and ergodicity of the jump-diffusion CIR process
TL;DR: In this paper, the jump-diffusion CIR process is studied and sufficient conditions on the Levy measure of the subordinator are provided under which the jump diffusion process is ergodic.
Journal ArticleDOI
Poisson Bandits of Evolving Shade of Gray
TL;DR: Stopping decisions in a model with Poisson bandits of "evolving shade of gray" are qualitatively different from those in optimal stopping or Poisson bandit models, and it is demonstrated that it may not be optimal to act immediately upon observation even if successes or failures are conclusive.
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Importance sampling and statistical Romberg method for Lévy processes
TL;DR: In this article, the authors proposed a statistical Romberg method to improve the complexity of the classical Monte Carlo method for simulatable compound Poisson processes and proved the central limit theorems of Lindeberg-Feller type for the inferred errors depending on the parameters.
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A solution selection problem with small symmetric stable perturbations
Franco Flandoli,Michael Högele +1 more
TL;DR: In this article, the zero-noise limit of differential equations with singular coefficients was investigated for the first time in the case when the noise is an α-stable process and it was proved that extremal solutions were selected and the respective probability of selection was computed.
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A quantum invitation to probability theory
TL;DR: Quantum probability theory and complex analysis for children is described in this article, with a focus on complex analysis of complex models. But it is not suitable for children with complex analysis.
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.