Parametric Estimation for Partially Hidden Diffusion Processes Sampled at Discrete Times

TL;DR: In this paper, a discrete sampling scheme was proposed to estimate finite dimensional parameters in both drift and diffusion coefficient under a continuous sampling scheme, where the sampling occurs at regularly spaced times intervals of length n = T such that n =T. The asymptotic normality of the estimators of parameters in drift was proved.

Abstract: A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$ is observed only when its path lies over some threshold $\tau$. On the basis of the observable part of the trajectory, the problem is to estimate finite dimensional parameter in both drift and diffusion coefficient under a discrete sampling scheme. It is assumed that the sampling occurs at regularly spaced times intervals of length $h_n$ such that $h_n\cdot n =T$. The asymptotic is considered as $T\to\infty$, $n\to\infty$, $n h_n^2\to 0$. Consistency and asymptotic normality for estimators of parameters in both drift and diffusion coefficient is proved.