  This paper proposes a widely applicable method of approximate
maximum-likelihood estimation for multivariate diffusion process from
discretely sampled data. A closed-form asymptotic expansion for transition
density is proposed and accompanied by an algorithm containing only basic and
explicit calculations for delivering any arbitrary order of the expansion. The
likelihood function is thus approximated explicitly and employed in statistical
estimation. The performance of our method is demonstrated by Monte Carlo
simulations from implementing several examples, which represent a wide range of
commonly used diffusion models. The convergence related to the expansion and
the estimation method are theoretically justified using the theory of Watanabe
[Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992)
139-159] on analysis of the generalized random variables under some standard
sufficient conditions.
