High-frequency Donsker theorems for Lévy measures

Abstract

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands.MSC 2000 subject classification: Primary: 60F05; Secondary: 60G51, 62G05

Efficient nonparametric inference for discretely observed compound Poisson processes

“…In the highfrequency regime, ∆ = ∆ n → 0 and n∆ n → ∞, the inverse nature of the problem vanishes as the number of observations grows. 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.…”

Quantile estimation for Lévy measures

“…However, both articles are restricted to Lévy processes with bounded variation and thus use only the first derivative of ψ. Focusing on settings that allow for parametric rates, the distribution function estimation have been considered by Nickl and Reiß [21] and Nickl et al [22] in the low and high-frequency regime, respectively.…”

“…The distribution function estimator  N h was studied by Nickl et al [22] in a high-frequency regime. For low frequency observations a modification of  N h was considered by Nickl and Reiß [21].…”

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

“…If one is interested in the entire jump measure, however, its estimation is rather difficult, at least in the presence of a diffusion component, as ∆ 1/2 n provides a natural bound to disentangle jumps from volatility. See Nickl et al (2016) and Hoffmann and Vetter (2016) for details in case of a Lévy process.…”

“…The situation is different with a view on the jump behaviour of the process, mainly for two reasons: There is much more flexibility in the choice of the jump measure than there is regarding the diffusive part, and even if one restricts the model to certain parametric families the standard situation is the one of β-stable processes, 0 < β < 2, which are quite difficult to deal with, at least in comparison to Brownian motion. To mention recent work besides the aforementioned monographs, see for example Nickl et al (2016) and Hoffmann and Vetter (2016) on the estimation of the jump distribution function of a Lévy process or Todorov (2015) on the estimation of the jump activity index from high-frequency observations. In the following, we are interested in the evolution of the jump behaviour over time in a completely non-parametric setting where we assume only stuctural conditions on the characteristic triplet of the underlying Itō semimartingale.…”