On the equivalence of financial structures with long-term assets

In a stochastic financial exchange economy, two financial structures are equivalent if, for each given state price, regardless the associated arbitrage free price, the marketable payoffs are identical. The key property of two equivalent financial structures is that, when associated with any standard exchange economy, they lead to the same financial equilibrium. We exhibit a sufficient condition for the equivalence of two financial structures with possibly long-term assets. We then apply this result to financial structures built upon primitive assets and their re-trading to get a necessary and sufficient condition. We also borrow an assumption from [7] to prove the equivalence between a financial structure and its reduced forms and between to complete financial structures.