Noise recovery for Lévy-driven CARMA processes and highfrequency behaviour of approximating Riemann sums

We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean Lévy processes. An L2-consistent es-timator for the increments of the driving Lévy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the high-frequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average models on a discrete grid. We compare their autocovariance structure with the one of sampled CARMA processes and show that the rule of integration plays a crucial role. Moreover, new insight into the kernel estimation procedure of Brockwell et al. (2012a) is given