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

Surprisingly, general estimators for nonlinear continuous time models based on stochastic differential equations are yet lacking. Most applications still use the Euler-Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as the Kessler, the Ozaki, or MCMC methods, lack a straightforward implementation and 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 also has an $L^p$ convergence rate of order 1, which was already known for LT. We prove under the less restrictive one-sided Lipschitz assumption that the estimators are consistent and asymptotically normal. A numerical study on the 3-dimensional stochastic Lorenz chaotic system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on both precision and computational speed compared to the state-of-the-art.