Parametric estimation for partially hidden diffusion processes sampled at discrete times

Abstract

For a one-dimensional diffusion process X = { X ( t ) ; 0 ≤ t ≤ 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 the estimation of a 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 → 0 , T → ∞ and n h n 2 → 0 as n → ∞ . Consistency and asymptotic normality for estimators of parameters in both drift and diffusion coefficients are proved.