Showing papers on "Hidden Markov model published in 1970"

Operator-Valued wide-sense Markov processes and solutions of infinite-dimensional linear differential systems driven by white noise

Abstract: Introduction. The study of finite linear stochastic differential systems with white-noise input has been the subject of several papers starting with Doob [4] (see also [2], [5], [7], [10], [14]). Some recent results of Falb [8] on infinitedimensional filtering depend on the structure of the solution of an infinitedimensional stochastic differential system. One also encounters an infinite stochastic differential system in the study of heat equations with random source (see [1] and [19]). All such processes arising from linear stochastic differential systems in both the finite-dimensional (see [2]), and infinite-dimensional (see [8]) cases, constitute appropriate generalizations of wide-sense Markov processes introduced by Doob [4]. We obtain in this paper some results on the structure of general wide-sense Markov processes which constitute generalizations of those in [14]. These generalizations constitute, in addition to the extension to the infinite-dimensional case, a sharpening of the finite-dimensional results of [14]. As applications of our results in the infinite-dimensional case, we present generalizations of the results of Beutler [2] and some results of Falb [8]. The approach of this paper is similar to that in [4] (stationary case) and [14] (for the non-stationary case), in that it depends on the structure of L2,M for an operator-valued measure M and the appropriate generalization of stochastic integrals. In view of some recent work of the authors [15], this approach seems extendible to the infinite-dimensional case. However, such an extension, although apparent, is not direct in view of the usual difficulties of working with infinite-dimensional processes; one of the difficulties is that the inverse of a covariance operator is unbounded, since, throughout this paper, Hilbert-Schmidt operator-valued stochastic processes as in [18] are considered. The paper is divided into six sections. After the preliminary Section 1, Section 2 contains the basic structure theorem of wide-sense Markov processes in terms of wide-sense martingales. In Section 4, we study the solutions of a