This paper studies what happens when traders make decisions with delayed or restricted information. In that situation the observed prices that traders use need not line up with the true price process. The authors ask when every payoff that depends only on the price history visible to traders can be exactly replicated. They show that the answer depends on which probability laws (called martingale measures) one allows for the projected price process.

The authors work in a simple, finite-time and finite-state setting. For a chosen probability measure, they form the optional projection of the discounted prices: that is the conditional expectation of the true prices given the traders’ information under that measure. A classical result (the first fundamental theorem under restricted information) says that absence of arbitrage is equivalent to the existence of a measure under which this projected price process has no predictable drift (is a martingale). Building on that, the paper proves a precise criterion for completeness: for a fixed projected process, every nonnegative payoff measurable with respect to the traders’ terminal price history is attainable exactly when all martingale measures for that projected process give the same values on the sigma-field generated by those prices. In plain terms, if every allowed probability law for the projected prices agrees about the relevant payoffs, then those payoffs can be hedged.

The work matters because real traders often act on delayed or partial information. The paper clarifies which payoffs can be priced and hedged in that setting. The authors give a finite-dimensional proof of their main criterion and keep a clear distinction between the information used for trading and the information that defines claims. They also study a concrete example: a binomial model where trading is delayed by k periods. Under the natural product martingale measure, the projected market is complete and its effective time horizon becomes (T−k)+, meaning you can only replicate payoffs that depend on prices after the delay.