Kalman--Bucy Filtering and Minimum Mean Square Estimator under Uncertainty

Abstract

In this paper, we study a generalized Kalman--Bucy filtering problem under uncertainty. The drift uncertainty for both signal process and observation process is considered, and the attitude to uncertainty is characterized by a convex operator (convex risk measure). The optimal filter or the minimum mean square estimator (MMSE) is calculated by solving the minimum mean square estimation problem under a convex operator. In the first part of this paper, this estimation problem is studied under $g$-expectation which is a special convex operator. For this case, we prove that there exists a worst-case prior $P^{\theta^{\ast}}$. Based on this $P^{\theta^{\ast}}$ we obtained the Kalman--Bucy filtering equation under $g$-expectation. In the second part of this paper, we study the minimum mean square estimation problem under general convex operators. The existence and uniqueness results of the MMSE are deduced.