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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.
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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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Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity
TL;DR: The first passage identities for one-dimensional stable processes with two-sided jumps were given in this paper, where the authors make use of path censoring and a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process.
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Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes
TL;DR: In this paper, a novel idea for a coupling of solutions of stochastic differential equations driven by Levy noise is presented, inspired by some results from the optimal transportation theory, and this coupling is used to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance.
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On magnitude, asymptotics and duration of drawdowns for L\'{e}vy models
TL;DR: In this paper, the authors consider the asymptotics of drawdown quantities when the threshold of the drawdown magnitude approaches zero and derive the law of duration of drawdowns for a large class of Levy processes (with a general spectrally negative part plus a positive compound Poisson structure).
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Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index
Aziz Issaka,Indranil Sengupta +1 more
TL;DR: In this article, a partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated, under appropriate assumptions for the first four cumulants of the driving subordinator, a Veceř-type theorem is proved.
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Spectral analysis of subordinate Brownian motions in half-line
TL;DR: In this paper, a generalized eigenfunction expansion of the transition operators of a one-dimensional Levy process with Levy-Khintchine exponent psi is derived, where psi is a complete Bernstein function.