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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
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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.
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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.
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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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Journal ArticleDOI
The convex minorant of a L\'{e}vy process
Jim Pitman,Gerónimo Uribe Bravo +1 more
TL;DR: In this article, a unified approach to the theory of convex minorants of L\'{e}vy processes with continuous distributions was proposed, and the Poisson-Dirichlet distribution of parameter 1 was shown to be the universal law of ranked lengths of excursions of a L\'evy process with continuous distribution over the interval $[0, 1].
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Scaling limits of anisotropic Hastings–Levitov clusters
TL;DR: In this paper, the authors consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic and give precise descriptions of the effects of the anisotropy.
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On the conservativeness and the recurrence of symmetric jump-diffusions
TL;DR: In this paper, sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process.
Journal ArticleDOI
General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula
Peter K. Friz,Atul Shekhar +1 more
TL;DR: In this paper, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes.
Journal ArticleDOI
A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part II: Discrete-Domain Theory
TL;DR: In this paper, an extended family of continuous-time autoregressive moving average (CARMA) processes that are solutions of stochastic differential equations driven by white Levy innovations is studied.