发表机构
Basis Research Institute; Cornell University; Massachusetts Institute of Technology（Basis研究所; 康奈尔大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对连续-离散非线性高斯滤波，提出维纳空间求积卡尔曼滤波器（WSC-KF），结合空间sigma点与维纳求积，以更高阶近似布朗运动迭代积分，实现优于或相当于标准方法的弱路径律近似。
AI 中文摘要
连续-离散滤波是信号处理和控制理论中的一个经典问题，其中驱动随机微分方程（SDE）的状态必须从带噪声的、部分离散时间观测中推断出来。相应的滤波算法通常被构造为离散时间方法并应用于SDE的离散化，这种方法受限于所产生的离散化误差，或者对一类高斯矩常微分方程（ODE）流进行建模，而后者对于非线性动力学是近似的。我们提出了一种新方法，即维纳空间求积卡尔曼滤波器（WSC-KF），用于这一连续-离散设置中的高斯滤波。WSC-KF将空间sigma点表示与维纳空间求积积分相结合，有效地将驱动布朗运动的迭代积分项近似到比标准离散化（例如，Ito-Taylor展开）或近似矩ODE流（例如，无迹卡尔曼滤波器）更高的阶。理论上，我们通过一个可调模型阶参数刻画了WSC-KF的路径律近似度。我们证明，根据所选择的模型阶，WSC-KF提供了与使用一阶Ito预测步的求积卡尔曼滤波器相当或更优的弱阶界。数值上，我们在一个全局Lipschitz的Levy面积扩散上确认了该阶。5度维纳求积匹配了现有连续-离散卡尔曼预测器忽略的混合迭代积分，同时保持高效可计算性；这是实现高阶弱路径律近似的机制。
英文摘要
Continuous-discrete filtering is a classical problem in signal processing and control theory, whereby the state driving a stochastic differential equation (SDE) must be inferred from noisy, partial discrete-time observations. Corresponding filtering algorithms are often formed as discrete-time methods and applied to a discretization of the SDE, an approach limited by the incurred discretization error, or model a type of Gaussian moment ordinary differential equation (ODE) flow, which is approximate for nonlinear dynamics. We introduce a new method, the Wiener-space cubature Kalman filter (WSC-KF), for Gaussian filtering in this continuous-discrete setting. The WSC-KF combines spatial sigma-point representations with Wiener space cubature integration, effectively approximating iterated integral terms of the driving Brownian motion to higher-order than standard discretization (e.g., Ito-Taylor expansions), or approximate moment-ODE flows (e.g., as in the unscented Kalman filter). Theoretically, we characterize the path-law approximation degree of WSC-KF through a tunable model order parameter. We prove that, depending on the model order chosen, WSC-KF provides weak order bounds comparable to or dominating that of a cubature Kalman filter using a first-order Ito predict step. Numerically, we confirm that rate on a globally Lipschitz Levy-area diffusion. Degree-5 Wiener cubature matches mixed iterated integrals that existing continuous-discrete Kalman predictors omit, while remaining efficiently computable; this is the mechanism enabling higher-order weak path-law approximation.
