A robust Kalman-Bucy filtering problem

TL;DR

This paper extends the classical Kalman-Bucy filtering framework to account for model uncertainty, reformulating the robust filtering problem as a classical problem under a new measure and deriving the optimal estimator.

Contribution

It introduces a robust filtering approach under model uncertainty, showing equivalence to a classical problem via Girsanov transformation and providing explicit equations for the optimal estimator.

Findings

Robust filtering problem is equivalent to a classical problem under a new measure.

Optimal estimator decomposes into classical estimator plus a model uncertainty term.

The approach handles uncertainty within the Kalman-Bucy framework.

Abstract

A generalized Kalman-Bucy model under model uncertainty and a corresponding robust problem are studied in this paper. We find that this robust problem is equivalent to an estimate problem under a sublinear operator. By Girsanov transformation and the minimax theorem, we prove that this problem can be reformulated as a classical Kalman-Bucy filtering problem under a new probability measure. The equation which governs the optimal estimator is obtained. Moreover, the optimal estimator can be decomposed into the classical optimal estimator and a term related to model uncertainty.