Consistent price systems for subfiltrations

Asymmetric or partial information in financial markets may be represented by different filtrations. We consider the case of a larger filtration F – the natural filtration of the “model world” – and a subfiltration $\hat{\mathcal F}$ that represents the information available to an agent in the “real world”. Given a price system on the larger filtration that is represented by a martingale measure Q and an associated numeraire S, we show that there is a canonical and nontrivial numeraire Ŝ such that the price system generated by (Ŝ,Q,$\hat{\mathcal F}$) is consistent, in a sense to be made precise, with the price system generated by (S,Q,F).