A new delta expansion for multivariate diffusions via the Itô-Taylor expansion

Abstract

In this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the Ito-Taylor expansion of the conditional expectation of the Dirac delta function. Our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models, particularly the time-inhomogeneous non-affine irreducible multivariate diffusions. We show that this new approach can be viewed as an extension of Ait-Sahalia (2002) and Lee et al. (2014) to the case of multivariate models. The derived expansions are proved to converge to the true probability density as the observational time interval shrinks. The obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data. Extensive numerical experiments demonstrate the accuracy and effectiveness of our approach.