Parametric estimation for partially hidden diffusion processes sampled at discrete times

Abstract

For a one dimensional diffusion process $X=\{X(t) ; 0\leq t \leq T \}$, we suppose that $X(t)$ is hidden if it is below some fixed and known threshold $τ$, 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 $h_n\to0$, $T\to\infty$ 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.