Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」We consider drift estimation of a discretely observed Ornstein-Uhlenbeck process driven by a possibly heavy-tailed symmetric Levy process with positive Blumenthal-Getoor activity index (BG index). Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being depending on the BG index and sampling frequency as well as the sample size. It turns out that the rate of convergence is determined by the most active part of the driving Levy process; the presence of a driving Wiener part leads to the rate familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Levy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Levy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions