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In this paper, we study a generalized Kalman-Bucy filtering problem underuncertainty. The drift uncertainty for both signal process and observationprocess is considered and the attitude to uncertainty is characterized by aconvex operator (convex risk measure). The optimal filter or the minimum meansquare estimator (MMSE) is calculated by solving the minimum mean squareestimation problem under a convex operator. In the first part of this paper,this estimation problem is studied under g-expectation which is a specialconvex operator. For this case, we prove that there exists a worst-case prior.Based on this worst-case prior we obtained the Kalman-Bucy filtering equationunder g-expectation. In the second part of this paper, we study the minimummean square estimation problem under general convex operators. The existenceand uniqueness results of the MMSE are deduced.

Kalman-Bucy filtering and minimum mean square estimator under uncertainty