Abstract
This paper gives a contribution to the physical interpretation of nonlinear filter obtained for continuous-time Markov processes governed by the stochastic (Ito) differential equations. Since the dynamical system for constructing the sample solution of stochastic differential equation is not physically available, results obtained for continuous-time Markov processes must be translated by known formula into results described by the ordinary differential equations. There is however a difficulty involved in such interpretation, since in most cases a truly optimal nonlinear filter i infinite dimensional. A method is presented here in this connection for finding the dynamical equation (in the ordinary sense) for the conditional probability density, from which a set of ordinary differential equations for conditional moments can be directly derived. The present nonlinear filter described by ordinary differential equations displays marked difference to the filter obtained by the finite dimensional approximation and the translation into a set of ordinary differential equations. Incidentally, the analysis shows that no serious problem is introduced whenever the linear observation scheme is available.
