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[t,+8), where t is a random time (but not an F-stopping time) satisfying Jacod's absolute continuity hypothesis, we compute the G-predictable compensator vG,X of the jump measure of X. Thanks to our theoretical results on vG,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 t. [ABSTRACT FROM AUTHOR]