ABSTRACT

I. SYSTEM AND OBSERVATION MECHANISMS We consider a simple example of system and observation mechanisms:

(2)

(5) w(t) is a Brownian motion process in R 1 with incremental covariance cr2 and e(t) is a finitely additive white noise in L2(T;R2) independent of w. If the observation noise is modelled by using a Brownian motion, the data are given by the “integral form” and are nowhere differentiable. This causes a serious difficulty when one handles real data. The remarkable advantage of the finitely additive white noise is that the results obtained are always in the form where real data can be directly used. In technical

The precise meaning of the above equations can be given by

where

terms we say that the results are always in robust form. (See [3] for more general information on finitely additive white noise theory.) The nonlinear filtering problem for general systems has also been studied and the related Zakai equation has been derived in [3].