Abstract
Closed-form density expansion is an effective method for estimating multivariate diffusion processes, such as the path-wise expansion method proposed by Li (2013) and the Hermite expansion method introduced in Yang et al. (2019) and Wan and Yang (2021). In this paper, we show that the formulas obtained from these two expansion methods are essentially the same. Utilizing explicit expressions for the conditional expectation of the multiplication of iterated Ito integrals, we provide an explicit expression for the path-wise expansion method based on Hermite polynomials for a special process and use it to recalculate the formulas for both methods, demonstrating their equivalence. We further show the equivalence result for general multivariate diffusion processes by introducing a quasi-Lamperti transform.
