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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
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Fundamental solution for super-critical non-symmetric Lévy-type operators
TL;DR: In this paper, the heat kernel was constructed for the case when the order of the operator is positive and smaller than 1 (but without excluding higher orders up to 2) under certain assumptions.
Journal ArticleDOI
Coupling Poisson Processes by Self-Decomposability
TL;DR: In this paper, a method to produce pairs of non-independent Poisson processes M(t), N(t) from positively correlated, self-decomposable, exponential renewals is presented.
Posted Content
Symbolic Calculus in Mathematical Statistics: A Review
TL;DR: In the last ten years, the use of symbolic methods has substantially extended both the theory and the applications of statistics and probability as mentioned in this paper, and the main tools are a formal generalization of the convolution of identical probability distributions, which allows us to employ compound Poisson random variables in various topics that are only somewhat interrelated.
Journal ArticleDOI
Finite-time survival probability and credit default swaps pricing under geometric Lévy markets
TL;DR: In this article, the first passage time over a fixed threshold for a pure-jump subordinator with negative drift was studied, and a closed-form formula for its survival function in terms of marginal density functions of the subordinator was derived.
DissertationDOI
Generalized Ornstein-Uhlenbeck Processes and Extensions
TL;DR: In this paper, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation for a bivariate generalized Ornstein-Uhlenbeck process were derived.
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.