Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process XX on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let PsemPsem be the set of all probability measures PP under which XX is a semimartingale. We construct processes (BP,C,νP)(BP,C,νP) which are jointly measurable in time, space, and the probability law PP, and are versions of the semimartingale characteristics of XX under PP for each P∈PsemP∈Psem. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.