Pathwise Representation of the Smoothing Distribution in Continuous-Time Linear Gaussian Models

We study the filtering and smoothing problem for continuous-time linear Gaussian systems. While classical approaches such as the Kalman-Bucy filter and the Rauch-Tung-Striebel (RTS) smoother provide recursive formulas for the conditional mean and covariance, we present a pathwise perspective that characterizes the smoothing error dynamics as an Ornstein-Uhlenbeck process. As an application, we show that standard filtering and smoothing equations can be uniformly derived as corollaries of our main theorem. In particular, we provide the first mathematically rigorous derivation of the Bryson-Frazier smoother in the continuous-time setting. Beyond offering a more transparent understanding of the smoothing distribution, our formulation enables pathwise sampling from it, which facilitates Monte Carlo methods for evaluating nonlinear functionals.