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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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Sparsity-Driven Statistical Inference for Inverse Problems
TL;DR: This thesis adopts a statistical perspective and model the signal as a realization of a stochastic process that exhibits sparsity as its central property, and proposes a novel nonlinear forward model and a corresponding algorithm for the quantitative estimation of the refractive index distribution of an object.
Journal ArticleDOI
Logconcave reward functions and optimal stopping rules of threshold form
TL;DR: In this paper, the authors explore the connection between increasing and logconcave reward functions and optimal stopping rules of threshold form, and show that if a reward function defined on Z is nonnegative, increasing, logcon-cave, then the optimal stopping rule is of threshold-form provided the underlying random walk is skip-free to the right.
Journal ArticleDOI
Properties of Poisson processes directed by compound Poisson-Gamma subordinators.
Khrystyna Buchak,Lyudmyla Sakhno +1 more
TL;DR: In this paper, the authors consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators and derive the expressions for their hitting times.
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Small ball probabilities for a class of time-changed self-similar processes
TL;DR: In this article, the authors established small-ball probabilities for a class of time-changed processes, where the inner process is a self-similar process and the outer process is an independent continuous process, each with a certain small ball probability.
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Asymptotics for first-passage times of L\'evy processes and random walks
D. Denisov,Vsevolod Shneer +1 more
TL;DR: In this article, the exact asymptotics for the distribution of the first time a L\'evy process $X_t$ crosses a negative level $-x was studied.
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.