Sensitivity of marketable payoffs with long-term assets

We consider a stochastic financial exchange economy with a finite dateevent tree representing time and uncertainty and a financial structure with possibly long-term assets. We exhibit a sufficient condition under which the set of marketable payoffs depends continuously on the arbitrage free asset prices. This generalizes previous results of Angeloni-Cornet and Magill-Quinzii involving only short-term assets. We also show that, under the same condition, the useless portfolios do not depend on the arbitrage free asset prices. We then derive an existence result for nominal assets for all state prices with assumptions only on the fundamental datas of the economy.