Statistical inference for time-changed Lévy processes via composite characteristic function estimation

Abstract

In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process Lt with independent components based on the low-frequency observations of the corresponding time-changed Lévy process $L_{\mathcal{T}(t)}$, where $\mathcal{T}$ is a nonnegative, nondecreasing real-valued process independent of Lt, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the Lévy density of Lt and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed Lévy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) Lévy processes.

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