Testing the characteristics of a Lévy process

Reiß, Markus

doi:10.1016/j.spa.2013.03.016

Cited by 16 publications

(31 citation statements)

References 25 publications

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“…If we assume not only stable-like jumps, but also that log-prices are given by a Lévy process, Reiß (2013) shows we can estimate β at the near-optimal rate n −β/4+ε , for any ε > 0. However, the assumption of Lévy behaviour is quite restrictive in a financial context, and unfortunately the approach of Reiß does not easily generalise to semimartingales.…”

Section: Near-optimal Estimation Of Jump Activitymentioning

confidence: 99%

Near-optimal estimation of jump activity in semimartingales

Bull¹

2016

Ann. Statist.

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In quantitative finance, we often model asset prices as semimartingales, with drift, diffusion and jump components. The jump activity index measures the strength of the jumps at high frequencies, and is of interest both in model selection and fitting, and in volatility estimation. In this paper, we give a novel estimate of the jump activity, together with corresponding confidence intervals. Our estimate improves upon previous work, achieving near-optimal rates of convergence, and good finite-sample performance in Monte-Carlo experiments.Comment: Published at http://dx.doi.org/10.1214/15-AOS1349 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Near-optimal Estimation Of Jump Activitymentioning

confidence: 99%

Near-optimal estimation of jump activity in semimartingales

Bull¹

2016

Ann. Statist.

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“…Consequently, Nickl, Reiß, Söhl and Trabs [30] prove a Donsker type of theorem on R for functionals of the Lévy measure for a much larger class of Lévy processes that may carry a diffusion component. They also derive goodness of fit tests complementing the work of Reiß [31] on testing Lévy processes. Previously Buchmann [4] showed a functional central limit theorem with an exponentially weighted sup-norm using continuous, although perhaps incomplete, observations.…”

Section: Introductionmentioning

confidence: 99%

Efficient nonparametric inference for discretely observed compound Poisson processes

Coca

2017

Probab. Theory Relat. Fields

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A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as confidence regions and tests are briefly discussed.MSC 2010 subject classification: Primary: 62G10; Secondary: 60F05, 60G51, 62G05, 62G15.

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“…The distribution of Y i is a convolution exponential and therefore not linear in ϑ. If Y i is more generally an increment of a Lévy process (L t ) t 0 , inference on the characteristic triplet of the Lévy process is a nonlinear problem since the dependence of the probability distribution of the marginals on the Lévy triplet is determined by the characteristic exponent, see the review by Reiß [29]. At the same time this model is of practical importance since Lévy processes are the main building blocks for mathematical modeling of stochastic processes.…”

Section: Introductionmentioning

confidence: 99%

Information bounds for inverse problems with application to deconvolution and Lévy models

Trabs¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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If a functional in a nonparametric inverse problem can be estimated with parametric rate, then the minimax rate gives no information about the ill-posedness of the problem. To have a more precise lower bound, we study semiparametric efficiency in the sense of Hájek-Le Cam for functional estimation in regular indirect models. These are characterized as models that can be locally approximated by a linear white noise model that is described by the generalized score operator. A convolution theorem for regular indirect models is proved. This applies to a large class of statistical inverse problems, which is illustrated for the prototypical white noise and deconvolution model. It is especially useful for nonlinear models. We discuss in detail a nonlinear model of deconvolution type where a Lévy process is observed at low frequency, concluding an information bound for the estimation of linear functionals of the jump measure.

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Testing the characteristics of a Lévy process

Cited by 16 publications

References 25 publications

Near-optimal estimation of jump activity in semimartingales

Near-optimal estimation of jump activity in semimartingales

Efficient nonparametric inference for discretely observed compound Poisson processes

Information bounds for inverse problems with application to deconvolution and Lévy models

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