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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.
Journal ArticleDOI
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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Risk Theory with Affine Dividend Payment Strategies
Hansjörg Albrecher,Arian Cani +1 more
TL;DR: In this article, a classical compound Poisson risk model with affine dividend payments is considered and closed-form expressions for the expected discounted dividends until ruin and the Laplace transform of the time to ruin can be derived for exponentially distributed claim amounts.
Journal ArticleDOI
Global and nonglobal solutions of a system of nonautonomous semilinear equations with ultracontractive Lévy generators
TL;DR: In this paper, a semilinear system of the form ∂ u i (t, x ) / ∂ t = k ( t ) A u i( t, x) + u i ′ β i ( t, x ), with Dirichlet boundary conditions on a bounded open set D ⊂ R d, where k is the infinitesimal generator of a symmetric Levy process Z ≡ { Z (t ) } t ≥ 0, β i > 1, i ∈ { 1, 2 } and i = 3 − i
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Time-inhomogeneous jump processes and variable order operators
TL;DR: In this article, the authors introduce non-decreasing jump processes with independent and time non-homogeneous increments, which generalize subordinators in the sense that their Laplace exponents are possibly different Bernstein functions for each time $t.
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On the characterisation of honest times that avoid all stopping times
TL;DR: In this paper, it was shown that a random time is an honest time that avoids all stopping times if and only if it coincides with the (last) time of maximum of a nonnegative local martingale with zero terminal value and no jumps while at its running supremum.