Predictable projections for point process filtrations

Summary We consider a point process with the Polish phase space (X,X) and a system of -fields (x),xX, generated by on certain sets (x)X. We define predictability for random processes indexed byX and for random measures onX and prove the existence and uniqueness of predictable and dual predictable projections under a regularity condition on . ForX=₂⁺ and under monotonicity assumptions on the sets x we will identify the predictable projections of some simple processes as regular versions of certain martingales.