Linear filtering system with arbitrary initial conditions

We consider linear filtering system with non-Gaussian initial condition. Two explicit and simple filtering formulae are obtained: one for the conditional density function of the estimated state and another for the conditional mean. Only 2n sufficient statistics need to be computed in real time, where n is the dimension of the state vector. It is shown that our formulae constitute natural extensions of the Kalman-Bucy filter. We also give an explicit solution to the derived Kolmogorov equation