Open Access
April 2024 Parameter estimation in nonlinear multivariate stochastic differential equations based on splitting schemes
Predrag Pilipovic, Adeline Samson, Susanne Ditlevsen
Author Affiliations +
Ann. Statist. 52(2): 842-867 (April 2024). DOI: 10.1214/24-AOS2371


The likelihood functions for discretely observed nonlinear continuous time models based on stochastic differential equations are not available except for a few cases. Various parameter estimation techniques have been proposed, each with advantages, disadvantages and limitations depending on the application. Most applications still use the Euler–Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as Kessler’s Gaussian approximation, Ozaki’s local linearization, Aït–Sahalia’s Hermite expansions or MCMC methods, might be complex to implement, do not scale well with increasing model dimension or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie–Trotter (LT) and the Strang (S) splitting schemes. We prove that S has Lp convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art.

Funding Statement

The European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska–Curie grant agreement No 956107, “Economic Policy in Complex Environments (EPOC)”; and Novo Nordisk Foundation NNF20OC0062958.
This work has been partially supported by MIAI@Grenoble Alpes, (ANR-19-P3IA-0003).


PP is also affiliated with the Bielefeld Graduate School of Economics and Management at the University of Bielefeld in Germany.

We would like to thank three anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the paper. We are thankful to the third reviewer for providing the HE method implementation for the Lorenz system.


Download Citation

Predrag Pilipovic. Adeline Samson. Susanne Ditlevsen. "Parameter estimation in nonlinear multivariate stochastic differential equations based on splitting schemes." Ann. Statist. 52 (2) 842 - 867, April 2024.


Received: 1 January 2023; Revised: 1 February 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2371

Primary: 62F12 , 62H12 , 62M99
Secondary: 37M15 , 60G65

Keywords: asymptotic normality , consistency , Lp convergence , splitting schemes , Stochastic differential equations , stochastic Lorenz system

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.52 • No. 2 • April 2024
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