Optimal nonlinear filtering for independent increment processes--Part II

The main result obtained in Part I [1]--which is contained in Theorem
1
of Part I--is generalized and extended in several ways. First, a new basic stochastic integro-partial differential equation for the conditional probability density function for the state of a nonlinear dynamic system with disturbance noise given noisy nonlinear measurements of the state is derived under the less restrictive assumption that the disturbance noise is an arbitrary independent increment process with an infinitely divisible distribution, and the measurement noise is a Gaussian independent increment process with an infinitely divisible distribution. It is then shown that, under proper restrictions, this basic equation reduces to either the Fokker-Planck equation for diffusion processes or the Kolmogorov-Feller equation for jump processes. Also, it is shown that this basic equation contains the earlier results of Kushner [2] and Wonhamt [3] as special cases. Next, it is shown how the result represented by this basic equation can be easily extended to include the case where the disturbance noise is a Markov process of the type initially assumed for the state of the dynamic system. Finally, it is shown that, in contrast to results obtained by Bryson and Johansen [4] and Cox [5] for the linear, Gaussian case, it is not generally possible to extend the result represented by this basic equation to include either the case where the measurement noise is a Markov process of the type initially assumed for the state of the dynamic system, or the case where the covariance matrix of the Gaussian measurement noise is singular. However, some incomplete results indicating when such extensions might be possible are given.