State estimation for partially observed jump processes

AbstractIn this paper partially observed jump processes are considered and optimal filtering equations are given for the conditional expectation of a functional on the past of the process.Rudemo [6] derived filtering equations for a partially observed jump Markov process. Snyder [3] gives equations for the conditional characteristic function of a jump process. Segall et al. [2] discuss filtering for processes with counting observations. Their work carries over to processes with counting observations the martingale methods that Fujisaki et al. [1] had used to derive nonlinear filtering equations for processes governed by Ito equations. Many further references to filtering for processes with discrete state measurements are given in the references cited.The objective of this paper is to show that by making use of the concept of a representation of a functional the idea of Rudemo's proof of [6, pp. 595–599] can be carried over to jump processes. The author feels that this is a very interesting proof because of its simplicity. It involves only calculations with conditional expectations and the rule for differentiation of a quotient