Generalized moment estimators for αdocumentclass[12pt]{minimal}usepackage{amsmath}usepackage{wasysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlength{oddsidemargin}{-69pt}egin{document}$$lpha $$end{document}-stable Ornstein–Uhlenbeck motions from discrete observations

摘要

We study the parameter estimation problem for discretely observed Ornstein–Uhlenbeck processes driven by αdocumentclass[12pt]{minimal}usepackage{amsmath}usepackage{wasysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlength{oddsidemargin}{-69pt}egin{document}$$lpha $$end{document}-stable Lévy motions. A method of moments via ergodic theory and via sample characteristic functions is proposed to estimate all the parameters involved in the Ornstein–Uhlenbeck processes. We obtain the strong consistency and asymptotic normality of the proposed joint estimators when the sample size n→∞documentclass[12pt]{minimal}usepackage{amsmath}usepackage{wasysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlength{oddsidemargin}{-69pt}egin{document}$$n ightarrow infty $$end{document} while the sampling time step h remains arbitrarily fixed. We also design a procedure to select the grid points in the characteristic functions in certain optimal way for the proposed estimators.