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

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. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2013, Vol. 41, No. 3, 1350-1380. This reprint differs from the original in pagination and typographic detail. 1 2 C. LI and the references given in Sørensen [59]. Taking the efficiency, feasibility and generality into account, maximum-likelihood estimation (hereafter MLE) can be a choice among others. However, for the increasingly complex real-world dynamics, likelihood functions (transition densities) are generally not known in closed-form and thus involve significant challenges in valuation. This leads to various methods of approximation and the resulting approximate MLE. The focus of this paper is to propose a widely applicable closed-form asymptotic expansion for transition density and thus to apply it in approximate MLE for multivariate diffusion process. 1.1. Background. To approximate likelihood functions, Yoshida [69] proposed to discretize continuous likelihood functions (see, e.g., Basawa and Prakasa Rao [13]); many others focused on direct approximation of likelihood functions (transition densities) for discretely monitored data, see surveys in, for example, Phillips and Yu [56], Jensen and Poulsen [36], Hurn, Jeisman and Lindsay [34] and the references therein. In particular, among various numerical methods, Lo [46] proposed to employ a numerical solution of Kolmogorov equation for transition density; Pedersen [55], Brandt and Santa-Clara [21], Durham and Gallant [25], Stramer and Yan [60], Beskos and Roberts [17], Beskos et al. [16], Beskos, Papaspiliopoulos and Roberts [15] and Elerian, Chib and Shephard [28] advocated the application of various Monte Carlo simulation methods; Yu and Phillips [73] developed an exact Gaussian method for models with a linear drift function; Jensen and Poulsen [36] resorted to the techniques of binomial trees. Since all these numerical methods are computationally demanding, real-world implementation has necessitated the development of analytical methods for efficiently approximating transition density. An adhoc approach is...

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