A special semimartingale derivation of smoothing and prediction equations (Q1113864)

This paper is concerned with the filtering, prediction and smoothing problems of a semimartingale \(\{\xi_ t\}\), given observations of a function of that process corrupted by additive noise. Let \({\mathcal F}_ t\) denote the \(\sigma\)-algebra generated by the observations between time 0 and time t. While the filtering problem consists in evaluating \(E(\xi_ t,{\mathcal F}_ t)\), the smoothing problem is concerned with \(E(\xi_ t/{\mathcal F}_ s)\) for \(t<s\), and the prediction problem with the same quantity for \(t>s.\) Let us point out a major difference between the results in the present paper and those in most of the existing literature: In the paper under review, evolution equations are obtained for \(E(\xi_ t,{\mathcal F}_ s)\) as a function of s, for t fixed, rather than the opposite.