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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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A class of scale mixtures of Gamma(k)-distributions that are generalized gamma convolutions
Anita Behme,Lennart Bondesson +1 more
TL;DR: In this article, an independent positive random variable with a density that is hyperbolically monotone (HIM) of order k has been defined, where k > 0 is an integer and Y is a standard Gamma(k) distributed random variable.
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Optimal Stopping for Strong Markov Processes : Explicit solutions and verification theorems for diffusions, multidimensional diffusions, and jump-processes.
TL;DR: In this article, the authors consider the optimal stopping problem of a strong Markov process, with a reward function and a discount rate, and find the stopping time such that the expected reward at the time of stopping is maximized.
Journal ArticleDOI
Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients
Arturo Kohatsu-Higa,Libo Li +1 more
TL;DR: In this paper, the authors used the backward parametrix method to prove the existence and regularity of the transition density associated to the solution process of a stable-like driven stochastic differential equation with Holder continuous coefficients.
Journal ArticleDOI
Reduced α-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise.
TL;DR: This study considers the stochastic averaging of systems where a linear CAM noise process in the infinite variance parameter regime drives a comparatively slow process and identifies the conditions required for the fast linear CAM process to have such an influence in driving a slower process and derives an (effectively) equivalent fast, infinite-variance process for which an existing stochastically averaging approximation is readily applied.
Journal ArticleDOI
Double-barrier first-passage times of jump-diffusion processes
TL;DR: An efficient and unbiased Monte-Carlo simulation is presented to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution, relevant in structural credit risk models if one considers two exit events.
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.