Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes

TL;DR

This paper introduces two new likelihood-based estimators for nonlinear multivariate stochastic differential equations using splitting schemes, demonstrating their efficiency, consistency, and superior performance in a complex 3D system.

Contribution

It proposes novel estimators based on Lie-Trotter and Strang splitting schemes, with proven convergence and efficiency, improving over existing methods in complexity and stability.

Findings

Strang splitting estimator has $L^p$ convergence rate of order 1.

Estimators are consistent and asymptotically efficient under less restrictive conditions.

Numerical results show the Strang estimator outperforms state-of-the-art methods in speed and accuracy.

Abstract

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, A\"it-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 $L^p$ 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.