高频区Lévy密度的无谱估计
Spectral-free estimation of Lévy densities in high-frequency regime
摘要：
我们构造了一个估计的Lévy密度的纯跳Lévy过程，可能是无限的变化，从离散观察的一个轨迹在高频率。我们的程序的新颖之处在于，我们直接估计依赖于路径策略的Lévy密度，而现有的程序依赖于谱技术。通过利用复合泊松近似，我们规避使用频谱技术，特别是Lévy-Khintchine公式。建立了一个线性小波估计器，并研究了其性能的$L_p$损失函数，$p\geq 1$，在Besov球。我们恢复经典的非参数速率有限变化的Lévy过程和一个大的非参数类的对称无限变化的Lévy过程。我们表明，该过程是强大的估计集接近临界值0时，也讨论了其存在的布朗部分的鲁棒性。
英文摘要：
We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy--Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of $L_p$ loss functions, $p\geq 1$, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a large nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.
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