Random time change and an integral representation for marked stopping times

Abstract

Consider the set $$C$$ of all possible distributions of triples (τ, κ, η), such that τ is a finite stopping time with associated mark κ in some fixed Polish space, while η is the compensator random measure of (τ, κ). We prove that $$C$$ is convex, and that the extreme points of $$C$$ are the distributions obtained when the underlying filtration is the one induced by (τ, κ). Moreover, every element of $$C$$ has a corresponding unique integral representation. The proof is based on the peculiar fact that EV τ, κ=0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT τ, κ isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of ζ, a suitably discounted version of η, 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 quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.