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 dow. We give necess ary and sufficient conditions for local absolute continuity between laws of such processes on a suitable canonical path space, and prove that likeliho od ratio processes do have a very simple form. Key tools are criteria for l ocal absolute continuity of laws of l-point motions-for arbitrary l-on stoc hastic flows, under an assumption on the structure of infinitesimal covaria nces.