The Dalang Morton Willinger Theorem under cone constraints

The Dalang–Morton–Willinger theorem [Stochastics Stochastic Rep. 29 (1990) 185] asserts, for a discrete-time perfect market model, that there is no arbitrage if and only if the discounted price process is a martingale with respect to an equivalent probability measure. The financial market is supposed to be perfect in the sense that there is no transaction cost, no imperfection on the numéraire, no short sale constraint, no constraint on the amounts invested, etc.In this note, we explore the same issue in the presence of such imperfections, more precisely, in the presence of polyhedral convex cone constraints. We first obtain a generalization of the Dalang–Morton–Willinger theorem [Stochastics Stochastic Rep. 29 (1990) 185]: we prove that under polyhedral convex cone constraints, absence of arbitrage is equivalent to the existence of a discount process such that, taking this process as a deflator, the net present value of any available investment opportunity is nonpositive.We then apply this general result to specific market imperfections fitting in the convex cone framework, like short sale constraints, solvability constraints, constraints on the quantities, amounts or proportions invested. We improve a result of Pham–Touzi [J. Math. Econ. 31 (2) (1999) 265]. We show that our model enables to deal with financial markets with possible imperfections on the numéraire (like different borrowing and lending rates, or more general convex cone constraints involving the numéraire).