Approximate Maximum Likelihood Estimation for One-Dimensional Diffusions Observed on a Fine Grid

Lu, Kevin W.; Paine, Phillip J.; Preston, Simon P.; Wood, Andrew T. A.

Abstract

We consider a one-dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô-Taylor expansion of the sample path, combined with an application of the so-called 𝜖-expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high-frequency sampling framework. This method of density approximation leads to a closed-form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood-based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate methods in the literature in terms of speed, accuracy, and ease of implementation.

Citation

Lu, K. W., Paine, P. J., Preston, S. P., & Wood, A. T. A. (2022). Approximate Maximum Likelihood Estimation for One-Dimensional Diffusions Observed on a Fine Grid. Scandinavian Journal of Statistics, 49(3), 1085-1114. https://doi.org/10.1111/sjos.12556

• Journal Article Type: Article
• Acceptance Date: Jul 20, 2021
• Online Publication Date: Sep 20, 2021
• Publication Date: 2022-09
• Deposit Date: Oct 8, 2021
• Publicly Available Date: Sep 21, 2022
• Journal: Scandinavian Journal of Statistics
• Print ISSN: 0303-6898
• Electronic ISSN: 1467-9469
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 49
• Issue: 3
• Pages: 1085-1114
• DOI: https://doi.org/10.1111/sjos.12556
• Keywords: Statistics, Probability and Uncertainty; Statistics and Probability
• Public URL: https://nottingham-repository.worktribe.com/output/6396707
• Publisher URL: https://onlinelibrary.wiley.com/doi/10.1111/sjos.12556
• Additional Information: This is the peer reviewed version of the following article: Lu, K. W., Paine, P. J., Preston, S. P., & Wood, A. T. A. (2021). Approximate maximum likelihood estimation for one-dimensional diffusions observed on a fine grid. Scand J Statist, which has been published in final form at https://doi.org/10.1111/sjos.12556. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.