A New Proof for the Linear Filtering and Smoothing Equations, and Asymptotic Expansion of Nonlinear Filtering

In this paper, we propose a new asymptotic expansion approach for nonlinear filtering based on a small parameter in the system noise. This method expresses the filtering distribution as a power series in the noise level, where the coefficients can be computed by solving a system of ordinary differential equations. As a result, it addresses the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. In the course of our derivation, we also show that classical linear filtering and smoothing equations, namely Kalman-Bucy filter and Rauch-Tung-Striebel smoother, can be obtained in a unified and transparent manner from an explicit formula for the conditional distribution of the hidden path.