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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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On Malliavin calculus and approximation of stochastic integrals for Lévy processes
TL;DR: In this paper, the Malliavin derivative operator for a Lévy process (Xt) was defined on the space D1,2 using a chaos expansion or in the case of a pure jump process also via an increment quotient operator.
Journal ArticleDOI
Stochastic Impulse Control with Regime-Switching Dynamics
TL;DR: In this article, the authors investigate dynamic optimal control problems that feature both intervention costs with a fixed component and partially controlled regime shifts and establish a direct approach to construct the value function and optimal strategies.
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Subexponential densities of compound Poisson sums and the supremum of a random walk
Takaaki Shimura,Toshiro Watanabe +1 more
TL;DR: In this article, it was shown that the class of all subexponential probability density functions on R_+$ is closed under generalized convolution roots of compound Poisson sums.
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Explicit formula for the supremum distribution of a spectrally negative stable process
TL;DR: In this article, the supremum distribution of a spectrally positive L\'evy process with infinite variation has been shown to be a generalization of the well-known formula for supremum distributions of Wiener processes.
Journal ArticleDOI
Parameter estimation and model testing for Markov processes via conditional characteristic functions
TL;DR: In this article, an empirical likelihood approach is proposed for parameter estimation and model specification testing, based on the conditional characteristic function for processes with either continuous or discontinuous sample paths, and a smoothed empirical likelihood ratio test for a parametric specification of the process is provided.
References
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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.
Book ChapterDOI
The Theory of Scale Functions for Spectrally Negative Lévy Processes
TL;DR: In this article, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative Levy processes, including the first extensive overview of how to work numerically with scale functions.
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.
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.