Explicit Pathwise Expansion for Multivariate Diffusions and Its Application to Equivalence of Density Expansions

25 Pages Posted: 12 Feb 2021 Last revised: 13 Mar 2024

See all articles by Nan Chen

Nan Chen

The Chinese University of Hong Kong (CUHK)

Xiangwei Wan

Shanghai Jiao Tong University - Antai College of Economics & Management

Nian Yang

Nanjing University - School of Business

Date Written: September 1, 2023

Abstract

In this paper, we provide expressions based on Hermite polynomials for the pathwise expansion method introduced by Watanabe (1987), Yoshida (1992b) and Li (2013). Our approach has two key innovations. First, we introduce a quasi-Lamperti transform that unitizes the process’ diffusion matrix at the initial time, as it corresponds to a multi-dimensional uncorrelated normal distribution, thus facilitating subsequent analysis. Second, by utilizing explicit expressions for the conditional expectation of the multiplication of iterated Itô integrals, we derive explicit formulas for the conditional expectation of the pathwise expansion of a general function on the transformed diffusion. Applying the newly derived method to the conditional expectation of the Dirac delta function and Hermite polynomial functions, respectively, we obtain alternative expressions for the pathwise based density expansion proposed by Li (2013) and the Hermite polynomials based density expansion introduced in Yang et al. (2019) and Wan and Yang (2021). We show that the formulas obtained from these two expansion methods are essentially the same by rearranging the terms according to the increasing order of Hermite polynomials.

Keywords: Pathwise Taylor Expansion, Hermite Expansion, Ito-Taylor Expansion, Equivalence, Transition densities, Multivariate diffusions

JEL Classification: C13, C32, G13, C63

Suggested Citation

Chen, Nan and Wan, Xiangwei and Yang, Nian, Explicit Pathwise Expansion for Multivariate Diffusions and Its Application to Equivalence of Density Expansions (September 1, 2023). Available at SSRN: https://ssrn.com/abstract=3748893 or http://dx.doi.org/10.2139/ssrn.3748893

Nan Chen

The Chinese University of Hong Kong (CUHK) ( email )

Shatin, N.T.
Hong Kong
Hong Kong

HOME PAGE: http://www.se.cuhk.edu.hk/people/nchen.html

Xiangwei Wan

Shanghai Jiao Tong University - Antai College of Economics & Management ( email )

No.1954 Huashan Road
Shanghai Jiao Tong University
Shanghai, Shanghai 200030
China
+86-21-52301570 (Phone)

HOME PAGE: http://sites.google.com/view/wanxiangwei/research

Nian Yang (Contact Author)

Nanjing University - School of Business ( email )

Nanjing
China

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