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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.
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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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Clustered continuous-time random walks: diffusion and relaxation consequences.
Karina Weron,Aleksander Stanislavsky,Agnieszka Jurlewicz,Mark M. Meerschaert,Hans-Peter Scheffler +4 more
TL;DR: This work presents a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times, and shows that the CTRW scaling limits are time-changed processes.
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Hedging electricity swaptions using partial integro-differential equations
TL;DR: In this article, the authors considered a general class of Hilbert space valued exponential jump-diffusion models, and derived the optimization problem for the quadratic hedging problem under the risk neutral measure and state a representation of its solution.
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L\'evy Processes and L\'evy White Noise as Tempered Distributions
Robert C. Dalang,Thomas Humeau +1 more
TL;DR: The authors showed that if the Levy measure associated with this noise has a positive absolute moment, then the Levy white noise almost surely takes values in the space of tempered distributions, and that the event on which the LSH is a tempered distribution has probability zero.
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Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes
TL;DR: In this article, the existence of viscosity solutions to non-linear integro-differential equations was studied using probabilistic methods, and new existence and uniqueness results were obtained for the Feller process for sublinear expectations and Feller processes on classical probability spaces.
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First passage times of Lévy processes over a one-sided moving boundary
TL;DR: In this paper, the authors studied the one-sided exit problem with a moving boundary for Levy processes and showed that if the boundary behaves asymptotically as t for some < 1/2, then the probability that the process stays below the boundary behaved as in the case of a constant boundary.