We consider the problem of estimating states and parameters in a model based on a system of coupled stochastic differential equations, based on noisy discrete-time data. Special attention is given to nonlinear dynamics and state-dependent diffusivity, where transition densities are not available in closed form. Our technique adds states between times of observations, approximates transition densities using, e.g., the Euler-Maruyama method and eliminates unobserved states using the Laplace approximation. Using case studies, we demonstrate that transition probabilities are well approximated, and that inference is computationally feasible. We discuss limitations and potential extensions of the method.

翻译：我们研究了基于含噪声离散时间数据，在耦合随机微分方程系统中估计状态与参数的问题。特别关注非线性动力学与状态依赖扩散率的情形，此类情形下转移密度通常无法以闭式表达。本方法通过在观测时刻之间插入状态变量，采用（例如）欧拉-丸山方法近似转移密度，并利用拉普拉斯近似消除未观测状态。通过案例研究，我们证明该方法能有效近似转移概率，且推断过程具有计算可行性。最后讨论了该方法的局限性及潜在扩展方向。