Statistical models for birth and death on a flow: Local absolute continuity and likelihood ratio processes

We consider statistical models for birth and death on a flow. We give necessary and sufficient conditions for local absolute continuity between laws of such processes on a suitable canonical path space, and prove that likelihood ratio processes do have a very simple form. Key tools are criteria for local absolute continuity of laws of l-point motions - for arbitrary l - on stochastic flows, under an assumption on the structure of infinitesimal covariances.