Properties of the maximum likelihood estimation of constant drift and diffusion parameters in $k$-dimensional diffusion process observed at discrete random sampling

Abstract

This article is concerned with the problem of a parameter estimation of the constant drift and diffusion coefficients: unknown vector $A$ and unknown positive definite matrix $B$, respectively, of a $k$-dimesional diffusion type process, when the observations at the moment of random point process are given. We compute the means and variances of the maximum likelihood estimators and establish their asymptotic properties. The unbiasedness, the strong consistemcy and the asymptotical efficiency of the estimation for $A$ are proved. The estimator of $B$ is unbiased and consistent and the variance of this estimator does not depend on the distribution of the random moments of observations.