Recent progress in stochastic processes--A survey

Abstract

This paper surveys some of the literature on applications of stochastic processes published during the period 196&1972. This survey highlights recent developments in the application of martingale theory. Because this subject is relatively new to the engineering community, a tutorial exposition of some aspects of the martingale calculus is also included.

A view of three decades of linear filtering theory

“…Despite these many successes, however, the use of such expansions is diminishing for several reasons. One is ,of course that K-L expansions really apply only to Gaussian (or second-order) processes, while recently martingale theory has enabled significant headway to be made with non-Gaussian processes (see, e.g., the survey [327] by The ease of solving the smoothing equation 151is heavily dependent upon the assumption that z(e) and v(a) are uncorrelated, which makes the right side equal to R,(t,s). Otherwise we would also have [cf.…”

“…The reasons for our attention go beyond the specific algorithm and are more broadly connected with the importance of dynamical structure in data-processing algorithms. Unfortunately communication engineers and information theorists have lagged behind control engineers in appreciating this fact, though, as pointed out in Wong's recent survey [327], the gap is closing. Kalman [64], [68], [69], changed the conventional formulation of the problem by giving, not the covariance of the signal process, but a "model" for it as the output of a dynamical linear system driven by white noise.…”

Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage

“…Cette propriete (10) de la mesure~t -previsible associee est en fait un outil extrernement puissant en theorie des systernes, Elle permet ce que Wong [104] appelle un 'calcul stochastique', c'est-a-dire un moyen de pousser les calculs sans trop reflechir. On a en effet ala fois la definition d'une 'integrale stochastique', ici l'integrale de tout processus~t -previsible par rapport aux martingales (5) A:(w)<oo; il est egalernent evident qu'il existe wEn;-n 1 et t<oo tels que A?…”

Estimation de l'État d'une file d'attente et du temps de panne d'une machine par la méthode de semi-martingales

“…Wong avait implicitement fait usage de ce principe dans [16] pour calculer Ie gain de filtres du type Kalman-Bucy. L'analogie formelle de ce resultat avec Ie principe de projection dans les sous-espaces de Hilbert de L 2(P) est anoter.…”