An anticipative linear filtering equation

In the classical Kalman–Bucy filter and in the subsequent literature so far, it has been assumed that the initial value of the signal process is independent of both the noise of the signal and of the noise of the observations. The purpose of this paper is to prove a filtering equation for a linear system where the (normally distributed) initial value X 0 of the signal process X t has a given correlation function with the noise (Brownian motion B t ) of the observation process Z t . This situation is of interest in applications to insider trading in finance. We prove a Riccati type equation for the mean square error S ( t ) : = E [ ( X t − X ˆ t ) 2 ] ; 0 ≤ t ≤ T , where X ˆ t is the filtered estimate for X t . Moreover, we establish a stochastic differential equation for X ˆ t based on S ( t ) . Our method is based on an enlargement of filtration technique, which allows us to put the anticipative linear filter problem into the context of a non-anticipative two-dimensional linear filter problem with a correlation between the signal noise and the observation noise.