Expansion of Filtrations

Abstract

By an expansion of the filtration F = (ℱ_t)_t≥0, we mean that we expand the filtration F to get a new filtration ℍ₌(H _t)_t≥0 which satisfies the usual hypotheses and F _t ⊂ H _t each t ≥ 0. There are three questions we wish to address: (1) when does a specific, given semimartingale remain a semi-martingale in the enlarged filtration; (2) when do all semimartingales remain semimartingales in the enlarged filtration; (3) what is a new decomposition of the semimartingale for the new filtration.