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

The observation process 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 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.