Random time change and an integral representation for marked stopping times (Q1123482)

Consider the set \({\mathcal C}\) of all possible distributions of triples (\(\tau\),\(\kappa\),\(\eta)\), such that \(\tau\) is a finite stopping time with associated mark \(\kappa\) in some fixed Polish space, while \(\eta\) is the compensator random measure of (\(\tau\),\(\kappa)\). We prove that \({\mathcal C}\) is convex, and that the extreme points of \({\mathcal C}\) are the distributions obtained when the underlying filtration is the one induced by (\(\tau\),\(\kappa)\). Moreover, every element of \({\mathcal C}\) has a corresponding unique integral representation. The proof is based on the peculiar fact that E \(V_{\tau,\kappa}=0\) for every predictable process V which satisfies a certain moment condition. From this it also follows that \(T_{\tau,\kappa}\) is U(0,1) whenever T is a predictable mapping into [0,1] such that the image of \(\zeta\), a suitably discounted version of \(\eta\), is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasi- leftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.