Market Completeness and Optional Projections under Restricted Information

In a finite discrete-time market, trading decisions may be predictable with respect to a filtration that does not adapt asset prices. The first fundamental theorem then characterizes absence of arbitrage by measures under which the optional projection of discounted prices is a martingale. We examine the corresponding completeness question. For a fixed projection, every claim measurable with respect to its terminal price history is attainable precisely when all equivalent martingale measures of that fixed process agree on the claim sigma-field. We give the finite-dimensional proof, retaining the distinction between the trading filtration and the claim sigma-field. If the projection is common to all optional martingale measures of the original prices, projected completeness implies uniqueness of their restrictions, but the converse fails even in a three-state model with a unique optional martingale measure. For the binomial model under its product martingale measure, a delay of $k$ periods yields a complete projected market with effective horizon $(T-k)^+$. An explicit replication construction and finite examples distinguish this completeness from replication at the original prices.