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This paper proposes a widely applicable method of approximatemaximum-likelihood estimation for multivariate diffusion process fromdiscretely sampled data. A closed-form asymptotic expansion for transitiondensity is proposed and accompanied by an algorithm containing only basic andexplicit calculations for delivering any arbitrary order of the expansion. Thelikelihood function is thus approximated explicitly and employed in statisticalestimation. The performance of our method is demonstrated by Monte Carlosimulations from implementing several examples, which represent a wide range ofcommonly used diffusion models. The convergence related to the expansion andthe 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 standardsufficient conditions.

Maximum-likelihood estimation for diffusion processes via closed-form density expansions