One Application of Generalized Stochastic Differential Equations to the Filtering Problem

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  • 一般化確率微分方程式の状態推定問題への応用

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In recent years there has been a great deal of activity in the study of estimation theory. It is the work of Kalman and Bucy that laid the foundation of this development. In Kalman-Bucy method, it is assumed that the sample paths of the message process and the observed process are continuous in t. Namely, it is supposed that the signal and the observation are generated by Ito's equations, more intuitively, the dynamical system and the observation system are corrupted by white noise. But in this paper, we assume that these systems are disturbed by constantly operating noise continuous in t and disturbances which cause random shocks or jumps in random time.<br>The object of this paper is to show that it is possible to apply the theory of generalized stochastic differential equations to studying dynamical systems corrupted by additive noise of which sample paths are not continuous in t but continuous in probability.<br>Also, it is very interesting to study qualitative properties of solutions of generalized stochastic differential equations. In fact, some efforts have been devoted to the discussion of the stability properties of the solution process and to the study of first exit times or, equivalently, to the problem of obtaining useful upper bounds to the probability that the state of the process will leave some given set at least once by a given time.<br>These investigations are closely related to the problem of stochastic controllability. However, this problem is fairly difficult and scarcely any result has ever been obtained. Therefore, the discussion in this paper is restricted to study only the optimal filtering problem based on Kalman-Bucy's theory. Consequently, this problem is reduced to solve the nonlinear differential equation governing the covariance matrix of the errors of the best linear estimate, called the matrix Aiccati equation and we try to integrate this fundamental equation approximately based on perturbation theory.

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