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: \mathrm{d}{X}_t^{ϵ}=\sqrt{ϵX_t^{ϵ}}\mathrm{d}{B}_t+\mathrm{d}{B}_{\mathit{t}}^{\mathit{\text{'}}}\sqrt{ϵX_t^{ϵ}+\rho I_m\mathrm{d}t}, 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 \{X_t^{ϵ}-X_t^{\mathrm{0}}/\sqrt{ϵh^{\mathrm{2}}(ϵ)},ϵ>\mathrm{0}\} satisfies a large deviation principle, and (X_t^{ϵ}-X_t^{\mathrm{0}})/\sqrt{ϵ} converges to a Gaussian process, where h(ϵ)\to +\infty and \sqrt{ϵ}h(ϵ)\to \mathrm{0} as ϵ\to \mathrm{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.

Keywords

Large deviation / moderate deviation / central limit theorem / Wishart process / eigenvalue

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Higher Education Press and Springer-Verlag Berlin Heidelberg