Moderate deviations and central limit theorem for small perturbation Wishart processes

Let Xε be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process Xε is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dX_t^\varepsilon = \sqrt {\varepsilon X_t^\varepsilon } dB_t + dB'_t \sqrt {\varepsilon X_t^\varepsilon } + \rho I_m dt$$\end{document}, X0 = x, where B is an m×m matrix valued Brownian motion and B′ denotes the transpose of the matrix B. In this paper, we prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {{{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right. \kern-\nulldelimiterspace} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right\}$$\end{document} satisfies a large deviation principle, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt \varepsilon }}} \right. \kern-\nulldelimiterspace} {\sqrt \varepsilon }}$$\end{document} converges to a Gaussian process, where h(ɛ) → +∞ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt \varepsilon h(\varepsilon ) \to 0$$\end{document} as ε → 0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of Xε are also obtained by the delta method.