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

Masahiro Kurisaki

TL;DR

This work provides a pathwise, probabilistically rigorous representation of the smoothing distribution in continuous-time linear Gaussian models by showing the smoothing error evolves as an Ornstein–Uhlenbeck process. The authors derive a general Gaussian conditional law for the hidden state with mean and covariance given by explicit kernel- and Riccati-based equations, from which the Kalman–Bucy filter, RTS smoother, and the Bryson–Frazier smoother follow as corollaries. A key advance is the explicit, pathwise description that enables direct sampling from the smoothing distribution, enhancing Monte Carlo evaluation of nonlinear path functionals and enabling refined estimation procedures. The framework also supports practical applications such as Monte Carlo EM, EKF-based linearizations, and simultaneous confidence bands for the hidden trajectory, strengthening both theoretical understanding and computational capabilities in continuous-time smoothing.

Abstract

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.

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

TL;DR

This work provides a pathwise, probabilistically rigorous representation of the smoothing distribution in continuous-time linear Gaussian models by showing the smoothing error evolves as an Ornstein–Uhlenbeck process. The authors derive a general Gaussian conditional law for the hidden state with mean and covariance given by explicit kernel- and Riccati-based equations, from which the Kalman–Bucy filter, RTS smoother, and the Bryson–Frazier smoother follow as corollaries. A key advance is the explicit, pathwise description that enables direct sampling from the smoothing distribution, enhancing Monte Carlo evaluation of nonlinear path functionals and enabling refined estimation procedures. The framework also supports practical applications such as Monte Carlo EM, EKF-based linearizations, and simultaneous confidence bands for the hidden trajectory, strengthening both theoretical understanding and computational capabilities in continuous-time smoothing.

Abstract

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.
Paper Structure (7 sections, 17 theorems, 125 equations)

This paper contains 7 sections, 17 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. One-dimensional Case
  3. Dynamics of the conditional process
  4. Filtering and smoothing equations
  5. Further Applications
  6. Lemma for (\ref{['eq-conditional-of-Z']})
  7. Proofs of Lemma \ref{['lemma-quadratic-form']} and (\ref{['eq-tensor-expectation']})

Key Result

Proposition 3.1

Let $T>0$ and define a probability measure $Q$ by $Q(A)=E\left[ 1_AZ_T^{-1} \right]$ for $A \in \mathcal{F}$, where Then, under the new measure $Q$, the process is a Brownian motion independent of $\{X_t\}_{0\leq t\leq T}$, and the law of $\{X_t\}_{0\leq t\leq T}$ is the same as its law under $P$. Furthermore, for $0\leq t \leq T$ and a random variable $U$ satisfying $E[|U|]<\infty$, it holds th

Theorems & Definitions (33)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['theorem-normal-expectation']}
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 23 more