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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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Asymptotic behavior of semistable L\'evy exponents and applications to fractal path properties
TL;DR: In this article, the authors proved sharp bounds on the tails of the Levy exponent of an operator semistable law on R^d, and applied these bounds to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-similar Levy processes.
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Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum
TL;DR: In this article, the authors revisited the classical results on the generalized St Petersburg sums and analyzed how the limit depends on the value of the maximum, and obtained an infinite sum representation of the distribution function of the possible semistable limits.
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Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise
TL;DR: It is observed that with further decreases of α to small values α≪1, with λ≠0, locking between blue and red may be restored, and this nonmonotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent.
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Non-local Conservation Law from Stochastic Particle Systems
Christian Olivera,Marielle Simon +1 more
TL;DR: In this paper, the authors consider an interacting particle system in the space variable of a system of stochastic differential equations driven by Levy processes and prove that the empirical density process converges uniformly to the solution of the $d$-dimensional fractal conservation law.
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Tail asymptotic of discounted aggregate claims with compound dependence under risky investment
TL;DR: In this paper, the authors considered the tail asymptotic of discounted aggregate claims with compound dependence under risky investment, where the price of risky investment was modeled by a geometric Levy process, while claims were modeled by one-sided linear process whose innovations further obeying a so-called upper tail independence.