Non-implementability of Arrow–Debreu equilibria by continuous trading under volatility uncertainty

In diffusion models, a few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow–Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow–Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow–Debreu equilibrium allocations are generically unattainable by dynamically trading a few long-lived assets.