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 Kessler's Gaussian approximation, Ozak's Local Linearization, A\"it-Sahalia's Hermite expansions, or MCMC methods, lack a straightforward implementation, 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.

翻译：令人惊讶的是，基于随机微分方程的非线性连续时间模型的一般估计方法目前仍缺乏。大多数应用仍在使用欧拉-丸山离散化方法，尽管已有许多证据表明其存在偏差。更复杂的方法，如凯斯勒的高斯近似、奥扎克的局部线性化、艾特-萨希亚的埃尔米特展开或马尔可夫链蒙特卡洛方法，要么缺乏直接的实现方式，要么随模型维度的增加可扩展性不佳，或可能出现数值不稳定性。我们提出了两种基于李-特罗特（LT）和斯特朗（S）分裂方案的高效且易于实现的似然估计方法。我们证明S方案具有$L^p$收敛阶为1的性质，而这一性质此前已知适用于LT方案。我们证明在较宽松的单侧利普希茨假设下，这些估计量是一致的且渐近有效的。针对三维随机洛伦兹系统的数值研究补充了我们的理论发现。模拟结果表明，在精度和计算速度方面，S估计量的表现优于现有最先进方法。