Estimating a class of diffusions from discrete observations via approximate maximum likelihood method

Abstract

ABSTRACTAn approximate maximum likelihood method of estimation of diffusion parameters based on discrete observations of a diffusion X along fixed time-interval and Euler approximation of integrals is analysed. We assume that X satisfies a stochastic differential equation (SDE) of form , with non-random initial condition. SDE is nonlinear in generally. Based on assumption that maximum likelihood estimator of the drift parameter based on continuous observation of a path over exists we prove that measurable estimator of the parameters obtained from discrete observations of X along by maximization of the approximate log-likelihood function exists, being consistent and asymptotically normal, and tends to zero with rate in probability when tends to zero with T fixed. The same holds in case of an ergodic diffusion when T goes to infinity in a way that goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of , and asymptotic efficiency of in this case.