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