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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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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.
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.
Journal ArticleDOI
Density and tails of unimodal convolution semigroups
TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
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.
References
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Proceedings Article
Nonconvex Penalization Using Laplace Exponents and Concave Conjugates
Zhihua Zhang,Bojun Tu +1 more
TL;DR: It is shown that the nonconvex logarithmic and exponential penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively, and the relationship between these two penalties is due to asymmetricity of the KL distance.
Posted Content
Extinction properties of multi-type continuous-state branching processes
TL;DR: In this article, the authors take a completely different approach and consider multi-type continuous-state branching processes, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms.
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The first passage event for sums of dependent L\'evy processes with applications to insurance risk
TL;DR: For the sum process of a bivariate L\'evy process with possibly dependent components, this article derived a quintuple law describing the first upward passage event caused by a jump.
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Heat content for stable processes in domains of $\R^d$
TL;DR: In this paper, the authors studied the small time behavior of the heat content of rotationally invariant stable processes in domains in R^d and showed that the behavior of heat content differs depending on the ranges $0<\alpha<1, $\alpha=1$ and $1 <\alpha\leq 2.
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Hitting times of points for symmetric Lévy processes with completely monotone jumps
TL;DR: In this article, small-space and large-time estimates and asymptotic expansion of the distribution function and derivatives of the density function of hitting times of points for symmetric Levy processes are studied.