Parametric estimation for partially hidden diffusion processes sampled at discrete times

For a one-dimensional diffusion process X={X(t);0≤t≤T}, we suppose that X(t)X(t) is hidden if it is below some fixed and known threshold ττ, 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 hnhn such that nhn=Tnhn=T. The asymptotic is when hn→0hn→0, T→∞T→∞ and nhn2→0 as n→∞n→∞. Consistency and asymptotic normality for estimators of parameters in both drift and diffusion coefficients are proved.