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Explicit pathwise expansion for multivariate diffusions with applications

Part of: Mathematical finance Markov processes Stochastic analysis

Published online by Cambridge University Press:  29 September 2025

Nan Chen ,
Xiangwei Wan  and
Nian Yang Open the ORCID record for Nian Yang [Opens in a new window]
Nan Chen*
Affiliation:
The Chinese University of Hong Kong
Xiangwei Wan*
Affiliation:
Shanghai Jiao Tong University
Nian Yang*
Affiliation:
Nanjing University
*
*Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Email: nchen@se.cuhk.edu.hk
**Postal address: Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China. Email: xwwan@sjtu.edu.cn
***Postal address: Department of Finance and Insurance, School of Business, Nanjing University, Nanjing, Jiangsu, China. Email: yangnian@nju.edu.cn
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Abstract

In this paper, we introduce a unified framework based on the pathwise expansion method to derive explicit recursive formulas for cumulative distribution functions, option prices, and transition densities in multivariate diffusion models. A key innovation of our approach is the introduction of the quasi-Lamperti transform, which normalizes the diffusion matrix at the initial time. This transformation facilitates expansions using uncorrelated Brownian motions, effectively reducing multivariate problems to one-dimensional computations. Consequently, both the analysis and the computation are significantly simplified. We also present two novel applications of the pathwise expansion method. Specifically, we employ the proposed framework to compute the value-at-risk for stock portfolios and to evaluate complex derivatives, such as forward-starting options. Our method has the flexibility to accommodate models with diverse features, including stochastic risk premiums, stochastic volatility, and nonaffine structures. Numerical experiments demonstrate the accuracy and computational efficiency of our approach. In addition, as a theoretical contribution, we establish an equivalence between the pathwise expansion method and the Hermite polynomial-based expansion method in the literature.

Keywords

Pathwise expansionquasi-Lamperti transformHermite expansionportfolio value-at-riskforward-starting option

MSC classification

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H35: Computational methods for stochastic equations
Secondary: 60J60: Diffusion processes 91G20: Derivative securities 91G70: Statistical methods, econometrics

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 58 , Issue 2 , June 2026 , pp. 592 - 625
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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