A Kalman filter for linear systems driven by time-space Brownian sheet

TL;DR

This paper extends the classical Kalman filter to linear systems driven by time-space Brownian sheets, deriving a stochastic integral equation for the conditional signal estimate, with illustrative examples.

Contribution

It introduces a novel time-space Kalman filter framework for systems driven by Brownian sheets, expanding filtering theory to multi-dimensional stochastic processes.

Findings

Derived a stochastic integral equation for the conditional signal

Extended Kalman filtering to time-space stochastic systems

Provided examples demonstrating the filtering approach

Abstract

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation, which can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations.