On Simulation of a Fractional Ornstein–Uhlenbeck Process of the Second Kind by the Circulant Embedding Method

We demonstrate how to utilize the Circulant Embedding Method (CEM) for simulation of fractional Ornstein–Uhlenbeck process of the second kind (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ fOU}_2$$\end{document}). The algorithm contains two major steps. First, the relevant covariance matrix is embedded into a circulant one. Second, a sample from the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ fOU}_2$$\end{document} is obtained by means of fast Fourier transform applied on the circulant extended matrix. The main goal of this paper is to explain both steps in detail. As a result, we obtain an accurate and an efficient algorithm for generating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ fOU}_2$$\end{document} random vectors. We also indicate that the above described procedure can be extended to applications with non-Gaussian marginals.