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

TL;DR

This paper introduces a novel method for approximate maximum-likelihood estimation of multivariate diffusion processes using a closed-form asymptotic expansion of transition densities, enabling explicit likelihood approximation from discretely sampled data.

Contribution

It provides a new explicit density expansion algorithm for diffusion processes, improving the accuracy and applicability of likelihood-based estimation methods.

Findings

The method performs well across various diffusion models.

The density expansion converges theoretically under standard conditions.

Monte Carlo simulations validate the effectiveness of the approach.

Abstract

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.