Infinite-Dimensional Filtering

Abstract

This paper presents a generalization of the standard Kalman–Bucy linear filtering problem to infinite dimensions. The infinite-dimensional linear stochastic dynamical system is represented as a stochastic evolution equation \[du(t,\omega ) = \mathcal{A}(t)u(t,\omega )dt + \mathcal{B}(t)dw(t,\omega ),\] where $A(t)$ is an unbounded operator, $\mathcal{B}(t)$ is a bounded operator, $w(t,\omega )$ is a Hilbert space-valued Wiener process and $u(t,\omega )$ is then a Hilbert space-valued stochastic process. The observation process is represented by \[dz(t,\omega ) = \mathcal{C}(t)u(t,\omega )dt + \mathcal{F}(t)dv(t,\omega ),\] where $\mathcal{C}(t)$ and $\mathcal{F}(t)$ are bounded operators and $v(t)$ is a finite-dimensional Wiener process. Using a combination of evolution equation techniques and abstract probability theory, the existence of an optimal filter for $u(t,\omega )$ based on the observation $z(t,\omega )$, $0 \leqq s \leqq t$, is established. As in the finite-dimensional theory, the filter may be obtained recursively by solving an infinite-dimensional Riccati equation. Similar results have been obtained by A. Bensoussan, Filtrage optimal des systems Lineares, 1971, where $\mathcal{A}(t)$ are specifically partial differential operators satisfying slightly stronger conditions. However, he also presents a theory for the case where $v(t)$ may be infinite-dimensional.