Discrete asset pricing under transaction costs and model uncertainty with and without short-sale constraints

We study discrete-time asset pricing with bid-ask spreads and model uncertainty. The family of probability measures enters the no-arbitrage condition through the union of its supports. In the single-period setting, we establish fundamental theorems of asset pricing with and without short-sale constraints. In the unconstrained market, no arbitrage is equivalent to the existence of a full-support martingale consistent price system. Under short-sale constraints, the martingale condition is replaced by a supermartingale condition. We then extend these results to a finite multi-period tree. The initial information is allowed to be nontrivial, so the initial trading cost and valuation bounds may depend on the initial state, and the corresponding inequalities are formulated conditionally. Finally, for a family of pricing measures, we introduce lower and upper robust supermartingale consistent price systems. We show that no arbitrage implies the existence of a lower system, while the existence of an upper system is sufficient for no arbitrage. A two-state example shows that the lower condition alone is not sufficient.