Strang 分裂估计器用于具有 Pearson 型乘法噪声的非线性多元随机微分方程
Strang splitting estimator for nonlinear multivariate stochastic differential equations with Pearson-type multiplicative noise
AI总结：
本文提出基于 Strang 分裂的参数估计器，用于具有 Pearson 型乘法噪声的非线性多元随机微分方程，通过分解随机系统为确定性非线性常微分方程和多元 Pearson 扩散，结合高斯过渡近似和精确矩，证明了估计器的一致性和渐近有效性。
摘要：
多元 Pearson 扩散由线性漂移和状态变量二次的扩散矩阵特征化。本文通过矩阵指数积分推导了该类的均值和协方差矩阵的闭式表达式，并扩展到更广泛的非线性扩散类，具有 Pearson 型乘法噪声。主要贡献是基于 Strang 分裂的新参数估计器，将随机系统分解为确定性非线性常微分方程和多元 Pearson 扩散。估计器通过组合其各自流并应用高斯过渡近似与 Pearson 组件的精确矩来构建。证明了估计器的一致性和渐近有效性。还引入了该类中的新模型，即 Student Kramers 振子，并证明了强解的存在性和不变测度的唯一性。通过模拟研究在该振子和多变量 Wright-Fisher 扩散上评估了估计器，其优于 Euler-Maruyama、高斯近似和局部线性化估计器。最后，应用 Student Kramers 振子到格陵兰冰芯数据。
英文摘要：
Multivariate Pearson diffusions are characterized by a linear drift and a diffusion matrix that is quadratic in the state variables. We derive closed-form expressions for the mean and covariance matrix of this class using matrix exponential integrals, and extend this framework to a broader class of nonlinear diffusions with Pearson-type multiplicative noise. The main contribution is a new parameter estimator for these nonlinear multiplicative models based on Strang splitting, which decomposes the stochastic system into a deterministic nonlinear ordinary differential equation and a multivariate Pearson diffusion. The estimator is constructed by composing their respective flows and applying a Gaussian transition approximation with exact moments from the Pearson component. We prove that the estimator is consistent and asymptotically efficient. We also introduce a new model within this class, the Student Kramers oscillator, and prove existence and uniqueness of the strong solution and of an invariant measure. We evaluate the estimator through simulation studies on this oscillator and on the multivariate Wright-Fisher diffusion from population genetics, where it outperforms the Euler-Maruyama, Gaussian approximation, and local linearization estimators. We conclude with an application to Greenland ice core data using the Student Kramers oscillator.
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