The reproducing kernel Hilbert spaces underlying linear SDE Estimation, Kalman filtering and their relation to optimal control

TL;DR

This paper explores the Hilbert space structures underlying linear stochastic differential equations, revealing new reproducing kernels for estimation and control, and providing a variational perspective on Kalman filtering and smoothing.

Contribution

It introduces explicit formulas for covariance kernels of Gaussian processes from linear SDEs and characterizes their RKHS, linking estimation and control through a unified Hilbert space framework.

Findings

Derived novel covariance kernels with minimal invertibility assumptions

Identified the RKHS structure of Gaussian processes in linear SDEs

Reproduced Kalman filter and smoother formulas via variational methods

Abstract

It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking extensive care to require minimal invertibility requirements on the operators, we give novel explicit formulas for these covariances. We also determine their reproducing kernel Hilbert spaces, stressing the symmetries between a space of forward-time trajectories and a space of backward-time information vectors. The two spaces play an analogue role for filtering to Sobolev spaces in variational analysis, and allow to recover the Kalman estimate through a direct variational argument. For comparison, we then recover the Kalman filter and smoother formulas through more classical arguments based on the innovation process. Extension to discrete-time observations or infinite-dimensional state, tough technical, would be straightforward.