Abstract
In this paper we consider a class of time-changed Lévy processes that can be represented in the form $Y_{s}=X_{\mathcal{T}(s)}$, where $X$ is a Lévy process and $\mathcal{T}$ is a non-negative and non-decreasing stochastic process independent of $X$. The aim of this work is to infer on the Blumenthal-Getoor index of the process $X$ from low-frequency observations of the time-changed Lévy process $Y$. We propose a consistent estimator for this index, derive the minimax rates of convergence and show that these rates can not be improved in general. The performance of the estimator is illustrated by numerical examples.
Highlights
Nonparametric statistical inference for Levy-type processes have been attracting the attention of researchers for many years starting from the works of Rubin and Tucker (1959) and Basawa and Brockwell (2007)
In this paper we consider a class of time-changed Levy processes that can be represented in the form Ys = XT (s), where X is a Levy process and T is a non-negative and non-decreasing stochastic process independent of X
We consider a class of processes known as the time-changed Levy process
Summary
Nonparametric statistical inference for Levy-type processes have been attracting the attention of researchers for many years starting from the works of Rubin and Tucker (1959) and Basawa and Brockwell (2007). An important remark is that the statistical analysis of the time-changed Levy models is much more difficult than the one of Levy models This lies in the fact that the increments of Y are not any longer independent and that neither the process X nor T is directly observable (see Belomestny, 2011). The question remains open whether the behaviour of ν at 0, expressed in terms of the BG index, can be recovered It follows from the results of this work that a consistent estimation of the BG index of X is basically possible, provided the Levy process X is independent of T and has a nonzero diffusion part. The unique similarity between this paper and Belomestny and Panov (2013) is the main idea to use the asymptotic behaviour of the characteristic function of the increments of the price processes to infer on the Blumenthal-Getoor (BG) index.
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