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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
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.
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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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Book ChapterDOI
Estimation and Calibration of Lévy Models via Fourier Methods
TL;DR: This chapter considers the estimation of the Levy triplet and the Blumenthal-Getoor index in Levy and time-changed Levy models and introduces a general spectral estimation/calibration approach that can be applied to these and many other statistical problems related to Levy processes.
Journal ArticleDOI
A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates
TL;DR: A Feynman–Kac representation of variational solutions to partial integro-differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes is derived and provides a rigorous basis for numerous applications in financial mathematics and in probability theory.
Journal ArticleDOI
Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes
Xu Sun,Jinqiao Duan +1 more
TL;DR: In this paper, the Fokker-Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian L\'evy processes.
Journal ArticleDOI
A complete Riemann zeta distribution and the Riemann hypothesis
TL;DR: In this paper, the Riemann hypothesis is proven to be true if and only if each σ(t) is a pretended-infinitely divisible characteristic function, which is defined in this paper, for each $1/2 1.
Journal ArticleDOI
A class of nonzero-sum investment and reinsurance games subject to systematic risks
TL;DR: In this paper, Liu et al. study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model, where each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims.