Filtering, Prediction and Smoothing for Counting Process Observations, a Martingale Approach

Abstract

In this paper we consider the filtering, prediction and smoothing problem for counting process observations. The method used is the theory of martingales and stochastic integrals. We define two equations in terms of martingales, forming the stochastic system. The observation equation associates the counting process with a rate process. The second equation expresses in general terms the evolution of the unobserved process that is to be estimated. We consider the filtering, prediction and smoothing problem using the least squares error criterion. The stochastic differential equations for the optimal estimates are derived. We discuss the resulting filter and give some examples.