Prediction and Filtering of Processes

Abstract

This chapter is devoted to a different class of applications complementing the preceding work. The first section contains a comparative analysis of general prediction operations relative to a convex loss function, and its relation to projection operators. This is refined in the next section, for least squares prediction with the Cramér-Hida method. Then Section 3 treats linear filters as formulated by Bochner [2]. The results are specialized and sharpened in Section 4 for linear Kalman-Bucy filters of interest in many applications. Then in Section 5, we consider nonlinear filtering, which is a counter part of the preceding showing that there are many new possibilities, as well as illustrating the essential use of the general theory of SDEs in this subject. Thus Sections 3–5 contain mathematical glimpses of some of the vast filter technology. Finally some related complements are included as exercises, often with sketches of proof.