Special solutions to some Kolmogorov equations arising from cubic sensor problems

1. Introduction. Ever since the technique of the Kalman-Bucy filter was pop- ularized, there has been an intense interest in developing nonlinear filtering theory. Basically we have a signal or state process x = {xt} which is usually not observable. What we can observe is a related process y = {yt}. The goal of nonlinear filtering is to determine the conditional expectation of the form E(�(xt) : ys,0 � st) whereis any C ∞ function or even better to compute the entire conditional probability density �(t,x) of xt given the observation history {ys : 0 � st}. In practical applications, it is preferable that the computation of conditional probability density be preformed recursively in terms of a statistic � = {�t}, which can be updated by using only the latest observations. In some cases, �t is computable with a finite system of differential equations driven by y. This leads to the ideal notion of finite dimensional recursive filter. By definition such a filter is a system: dt = �(�t)dt + p X i=1 �i(�t)dyit