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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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Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise
TL;DR: For a given bivariate Levy process (Ut, Lt)t = 0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analyzed in this paper, where the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described.
Posted Content
A hierarchical Archimedean copula for portfolio credit risk modelling
TL;DR: In this article, a hierarchical model of tail dependent asset returns is introduced for measuring portfolio credit risk within the structural framework, which allows for lower tail dependence, resulting in a more conservative credit risk assessment than a comparable Gaussian model.
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Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels.
TL;DR: In this article, the authors introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator.
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Optimal Importance Sampling Parameter Search for Lévy Processes via Stochastic Approximation
TL;DR: The author proposes stochastic approximation methods of finding the optimal measure change by the exponential tilting for Levy processes in Monte Carlo importance sampling variance reduction by either a constrained or unconstrained algorithm.
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Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations
TL;DR: In this article, the authors derived continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion and showed that these estimates are optimal in the BV-framework.