Predictable projections for point process filtrations (Q1333574)

We consider a point process \(\Phi\) with the Polish phase space \(({\mathbf X}, {\mathfrak X})\) and a system of \(\sigma\)-fields \({\mathcal F} (x)\), \(x \in {\mathbf X}\) generated by \(\Phi\) on certain sets \(\Gamma (x) \in {\mathfrak X}\). We define predictability for random processes indexed by \({\mathbf X}\) and for random measures on \({\mathbf X}\) and prove the existence and uniqueness of predictable and dual predictable projections under a regularity condition on \(\Phi\). For \({\mathbf X} = \mathbb{R}^ 2_ +\) and under monotonicity assumptions on the sets \(\Gamma_ x\) we will identify the predictable projections of some simple processes as regular versions of certain martingales.