On the Mayne-Fraser smoothing formula and stochastic realization theory for nonstationary linear stochastic systems

This paper is a shortened version of [1], its basic purpose being to provide an easily accessible introduction to the results of [1], many of which are presented here without proofs. However, we have tried to rearrange the material of [1], changing the logical order in which various topics are introduced, and occasionally we regard the results from a somewhat different angle. This has been done to increase the present paper´s usefulness as a complement to [1]. The work reported here is aimed at providing a theory of smoothing in the context of stochastic realization theory. This approach enables us to obtain stochastic interpretations of many important smoothing formulas and to explain the relationship between them. In this paper, however, we shall only consider one such formula, namely the Mayne-Fraser two-filter formula, which has a very natural interpretation in the stochastic realization setting; we refer the reader to [1] for further results. As a by-product, we also obtain certain results on the stochastic realization problem itself.