论文标题
基于分裂方案的非线性多元随机微分方程中的参数估计
Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes
论文作者
论文摘要
除几种情况外，基于随机微分方程的非线性连续时模型的离散观察到的非线性连续时模型的可能性函数。已经提出了各种参数估计技术，每种技术都有优点，缺点和限制，具体取决于应用程序。尽管有很多证据表明其偏见，但大多数应用程序仍使用Euler-Maruyama离散化。更复杂的方法，例如Kessler的高斯近似，Ozaki的局部线性化，Aït-Sahalia的Hermite扩展或MCMC方法可能很复杂，可以实现，并且不能随着模型维度的增加而进行良好的扩展，或者可以在数值上不稳定。我们提出了两个基于Lie-Trotter（LT）和Strang（S）分裂方案的两个高效且易于实现的估计器。我们证明S有$ l^p $收敛率1，该订单是LT已知的属性。我们表明，在限制性较小的单方面Lipschitz假设下，估计器是一致且渐近的效率。关于三维随机洛伦兹系统的数值研究补充了我们的理论发现。该仿真表明，与最先进的估计值相比，按精度和计算速度测量时，S估计器的表现最佳。
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 $L^p$ 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.
