Parametric estimation for partially hidden diffusion processes sampled at discrete times

Abstract

For a one dimensional diffusion process $X=\{X(t);0\le t\le T\}$, we suppose that $X(t)$ is hidden if it is below some fixed and known threshold $\tau$, but otherwise it is visible. This means a partially hidden diffusion process. The problem treated in this paper is to estimate finite dimensional parameter in both drift and diffusion coefficients under a partially hidden diffusion process obtained by a discrete sampling scheme. It is assumed that the sampling occurs at regularly spaced time intervals of length $h_n$ such that $n{h}_n=T$. The asymptotic is when $h_n\to 0$, $T\to \infty$ and $n{h}_n^2\to 0$ as $n\to \infty$. Consistency and asymptotic normality for estimators of parameters in both drift and diffusion coefficients are proved.