Structures and Algorithms in Stochastic Realization Theory and the Smoothing Problem

摘要

This report contains two main topics, each of which is connected to the stochastic realization problem. First, it considers some structural and algorithmic problems in wide sense stochastic realization theory which also have applicability to many problems outside the realm of stochastic realization theory but are here formulated in that framework. It considers some geometric questions concerning the solution set of the positive real lemma and provide a Hamiltonian framework for the non-Riccati algorithms of Kailath and Lindquist; these are then applied to the stochastic realization problem. Secondly, it applies the basic techniques and concepts of the strict sense (proper) stochastic realization theory of Lindquist and Picci and Ruckebusch to the discrete-time smoothing problem. This provides a natural interpretation of the Mayne-Fraser two-point formula as well as many other smoothing results, the interpretations of which have hitherto been quite unclear from a probabilistic point of view. Hence we have laid the ground work for a theory of smoothing which has so far been lacking.