Moderate deviations and central limit theorem for small perturbation Wishart processes

Abstract

Let X ^ε be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X ^ε is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: $dX_t^\varepsilon = \sqrt {\varepsilon X_t^\varepsilon } dB_t + dB'_t \sqrt {\varepsilon X_t^\varepsilon } + \rho I_m dt$, X ₀ = 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 $\left\{ {{{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right. \kern-\nulldelimiterspace} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right\}$ satisfies a large deviation principle, and ${{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt \varepsilon }}} \right. \kern-\nulldelimiterspace} {\sqrt \varepsilon }}$ converges to a Gaussian process, where h(ɛ) → +∞ and $\sqrt \varepsilon h(\varepsilon ) \to 0$ as ε → 0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X ^ε are also obtained by the delta method.