1,242 research outputs found
Asymptotic equivalence of jumps LĂ©vy processes and their discrete counterpart
Shorter version focusing on the statistical analysis of the LĂ©vy measure. A new example has been added.We establish the global asymptotic equivalence between a pure jumps LĂ©vy process on the time interval with unknown LĂ©vy measure belonging to a non-parametric class and the observation of Poisson independent random variables with parameters linked with the LĂ©vy measure . The equivalence result is asymptotic as tends to infinity. The time is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing
Estimation of the characteristics of a LĂ©vy process observed at arbitrary frequency
A Lévy process is observed at time points of distance Δ until time T. We construct an estimator of the Lévy-Khinchine characteristics of the process and derive optimal rates of convergence simultaneously in T and Δ. Thereby, we encompass the usual low- and high-frequency assumptions and obtain also asymptotics in the mid-frequency regime.Jump process, Lévy measure, deconvolution problem, statistical inverse problem
Decompositions of infinitely divisible nonnegative processes
International audienceWe establish decomposition formulas for nonnegative infinitely divisible processes. They allow to give an explicit expression of their LĂ©vy measure. In the special case of infinitely divisible permanental processes, one of these decompositions represents a new isomorphism theorem involving the local time process of a transient Markov process. We obtain in this case the expression of the LĂ©vy measure of the total local time process which is in itself a new result on the local time process. Finally, we identify a determining property of the local times for their connection with permanental processes
LIBOR additive model calibration to swaptions markets
In the current paper, we introduce a new calibration methodology for the LIBOR market model
driven by LIBOR additive processes based in an inverse problem. This problem can be splitted
in the calibration of the continuous and discontinuous part, linking each part of the problem
with at-the-money and in/out -of -the-money swaption volatilies. The continuous part is based
on a semidefinite programming (convex) problem, with constraints in terms of variability or
robustness, and the calibration of the LĂ©vy measure is proposed to calibrate inverting the
Fourier Transform
The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
Ito's construction of Markovian solutions to stochastic equations driven by a
LĂ©vy noise is extended to nonlinear distribution dependent integrands aiming at
the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the LĂ©vy-Khintchine type) with
variable coeffcients (diffusion, drift and LĂ©vy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or
nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov
processes of a long known fact for ordinary diffusions
On A New Class of Tempered Stable Distributions: Moments and Regular Variation
We extend the class of tempered stable distributions first introduced in
Rosinski 2007. Our new class allows for more structure and more variety of tail
behaviors. We discuss various subclasses and the relation between them. To
characterize the possible tails we give detailed results about finiteness of
various moments. We also give necessary and sufficient conditions for the tails
to be regularly varying. This last part allows us to characterize the domain of
attraction to which a particular tempered stable distribution belongs
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