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On infinitely divisible distributions with polynomially decaying characteristic functions
TLDR
In this article, necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function φ decays polynomially were provided. And they showed that φ is equivalent to φ being a Fourier multiplier on Besov spaces.Abstract:
We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $\phi$ decays polynomially. Under a mild regularity condition this polynomial decay is equivalent to $1/\phi$ being a Fourier multiplier on Besov spaces.read more
Citations
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Journal ArticleDOI
Quantile estimation for Lévy measures
TL;DR: In this article, the generalized quantiles of a Levy process are derived by the smallest values such that a negative jump larger than q τ + or a negative hop smaller than − q τ −, respectively, is expected only once in 1 / τ time units.
DissertationDOI
Infinitely divisible and related distributions and Lévy driven stochastic partial differential equations
TL;DR: In this paper, the authors studied the class of quasi-infinitely divisible distributions and Lévy driven stochastic partial differential equations and obtained bounds of the integral modulus of continuity in terms of the characteristic triplet.
Journal ArticleDOI
Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps
TL;DR: In this article, a fine analysis of a Gaussian approximation for the small jumps of Levy processes with infinite Levy measure in total variation distance is presented and extensively discussed, and new upper bounds for the total variation distances between discrete observations of Levy process are provided.
Posted Content
On the integral modulus of infinitely divisible distributions
TL;DR: In this paper, the integral modulus of continuity of probability densities of infinitely divisible distributions is derived for the case of random integrals with respect to a L'Evy process.
Journal ArticleDOI
On the integral modulus of continuity of infinitely divisible distributions, especially of stochastic integrals
TL;DR: In this paper, the integral modulus of continuity of probability densities of infinitely divisible distributions is derived for random integrals with respect to a Levy process, and major differences between integrals over compact and non-compact intervals are shown.
References
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Book
Theory of function spaces
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.
Book
Lévy processes and infinitely divisible distributions
TL;DR: 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.
Journal ArticleDOI
On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems
TL;DR: In this paper, it was shown that the difficulty of deconvolution depends on the smoothness of error distributions: the smoother, the harder it is to estimate the density of a random variable.
Journal ArticleDOI
Operator–valued Fourier multiplier theorems on Besov spaces
TL;DR: In this article, a general Fourier multiplier theorem for operator-valued multiplier functions on vector-valued Besov spaces is presented, where the required smoothness of the multiplier functions depends on the geometry of the underlying Banach space.
Journal ArticleDOI
Spectral calibration of exponential Lévy models
Denis Belomestny,Markus Reiß +1 more
TL;DR: It is shown that this inverse problem of calibrating an exponential Lévy model based on market prices of vanilla options is in general severely ill-posed and the exact minimax rates of convergence are derived.