On the compensator of step processes in progressively enlarged filtrations and related control problems

For a step process X with respect to its natural filtration F, we denote by G the smallest right-continuous filtration containing F and such that another step process H is adapted. We investigate some structural properties of the step process X in G. We show that Z = (X,H) possesses the weak representation property with respect to G. Moreover, in the case H = 1_[τ,+∞), where τ is a random time (but not an F-stopping time) satisfying Jacod’s absolute continuity hypothesis, we compute the G-predictable compensator ν^{G,X} of the jump measure of X. Thanks to our theoretical results on ν^{G,X}, we can consider stochastic control problems related to model uncertainty on the intensity measure of X, also in presence of an external risk source modeled by the random time τ.