Multivariate Pearson diffusions, also known as polynomial diffusions, are characterized by a linear drift vector and a diffusion matrix that is quadratic in the state variables. We derive exact closed-form expressions for the mean and covariance matrix of this class by using results on matrix exponential integrals. We then extend this framework to a broader class of nonlinear diffusions with Pearson-type multiplicative noise. The main contribution of this paper is a new parameter estimator for these nonlinear models based on Strang splitting (SS). The proposed method decomposes the stochastic system into a deterministic nonlinear ordinary differential equation (ODE) and a multivariate Pearson diffusion. We construct the SS estimator by composing their respective flows and applying a Gaussian transition approximation parameterized by the exact moments of the Pearson component. We prove that the SS estimator is consistent and asymptotically efficient. Furthermore, we introduce a new model within this broader class, which we call the Student Kramers oscillator, and we prove existence and uniqueness of the strong solution as well as existence of an invariant measure. We evaluate the SS estimator through simulation studies on this new oscillator and on the multivariate Wright-Fisher diffusion from population genetics. These simulations demonstrate that the SS estimator outperforms the standard Euler-Maruyama estimator, the Gaussian approximation estimator, and the local linearization estimator. Finally, we apply the SS estimator to fit the Student Kramers oscillator to Greenland ice core data.

翻译：多元Pearson扩散（亦称多项式扩散）的特征在于线性漂移向量以及关于状态变量的二次型扩散矩阵。我们利用矩阵指数积分的结果，导出了该类扩散的均值和协方差矩阵的精确闭式表达式。随后，我们将该框架推广至更广泛的含Pearson型乘性噪声的非线性扩散类。本文的主要贡献在于提出了一种基于Strang分裂（SS）的针对这些非线性模型的新参数估计量。该方法将随机系统分解为一个确定性非线性常微分方程（ODE）和一个多元Pearson扩散。通过组合各自的流并应用以Pearson分量的精确矩为参数的高斯转移近似，我们构建了SS估计量。我们证明了SS估计量具有相合性和渐近有效性。此外，我们在这一更广泛的模型类中引入了一个新模型，称之为Student Kramers振子，并证明了其强解的存在唯一性以及不变测度的存在性。我们通过对该新振子以及来自群体遗传学的多元Wright-Fisher扩散的模拟研究来评估SS估计量。这些模拟表明，SS估计量的性能优于标准的Euler-Maruyama估计量、高斯近似估计量和局部线性化估计量。最后，我们将SS估计量应用于拟合Student Kramers振子与格陵兰冰芯数据。