# A filtering problem with uncertainty in observation

## TL;DR

This paper develops a robust filtering approach for a generalized Kalman-Bucy model under uncertainty, deriving an optimal estimator using advanced stochastic analysis techniques.

## Contribution

It introduces a novel robust filtering framework under model uncertainty and derives the minimum mean square estimator using Girsanov and minimax theorems.

## Key findings

- Derived the equivalence between robust filtering and sublinear operator estimation
- Obtained the optimal estimator for the signal process under uncertainty
- Extended classical Kalman filtering to uncertain model settings

## Abstract

This paper is concerned with a generalized Kalman-Bucy filtering model and corresponding robust problem under model uncertainty. We find that this robust problem is equivalent to considering an estimate problem under some sublinear operator. Therefore, we turn to obtaining the minimum mean square estimator under a sublinear operator. By Girsanov theorem and minimax theorem, we obtain the optimal estimator $\hat{x}_{t}$ of the signal process $x_{t}$ for given time $t\in\lbrack0,T]$.