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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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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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Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution
TL;DR: In this paper, a Markov chain Monte Carlo (MCMCMC) method was introduced and a reversible proposal kernel was designed to have a heavy-tailed invariant probability distribution. And the authors showed that their algorithm has a much higher convergence rate than the pre-conditioned Crank-Nicolson (pCN) algorithm and the random walk Metropolis algorithm.
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Wavelet Statistics of Sparse and Self-Similar Images
TL;DR: It is proved that spatially dilated versions of self-similar sparse processes are asymptotically Gaussian as the dilation factor increases, and it is shown that the coarse-scale wavelet coefficients of these processes are also asymptonally Gaussian, provided the wavelet has enough vanishing moments.
Posted Content
Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials
TL;DR: In this article, a unified approach to solving jump-diffusion partial-integro-differential equations (PIDEs) was proposed, which is based on a second-order operator splitting on financial processes.
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Magnetic energies and Feynman-Kac-It\^o formulas for symmetric Markov processes
TL;DR: In this article, the authors considered related bilinear forms that generalize the energy form for a particle in an electromagnetic field, and obtained one by semigroup approximation and another, closed one by using a Feynman-Kac-It\^o formula.
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On two multistable extensions of stable Lévy motion and their semimartingale representation
TL;DR: In this article, the authors compare two definitions of multistable Levy motions, which are extensions of classical Levy motions where the stability index is allowed to vary in time, and show that one is a pure-jump Markov process, while the other one satisfies neither of these properties.