Estimation and decision for observations derived from martingales: Part I, Representations

The observation process
y
considered is an additive composition of continuous and discontinuous components. The additive Gaussian, point, and jump process models, treated separately in the past, are all included here simultaneously. Representations for
y
in terms of its innovations and following a Girsanov-type measure transformation are derived. These are then used to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.