The modeling of randomly modulated jump processes

We treat the problem of modeling events happening randomly in time at a rate determined by some other random "signal" process. It is shown that a martingale model includes all previously proposed models for such problems and also covers the difficult case of "past-dependent" signals that arises in feedback communication and control problems. It turns out that this model is very similar to the well-known signal-in-additive-white-Gaussian-noise model, so that it can be used conveniently in solving problems of detection of signals in jump processes and estimation of signals from jump processes.