Arbitrage and state price deflators in a general intertemporal framework

In securities markets, the characterization of the absence of arbitrage by the existence of state price deflators is generally obtained through the use of the Kreps-Yan theorem. This paper deals with the validity of this theorem (see Kreps, D.M., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15-35; Yan, J.A., 1980. Caracterisation d'une classe d'ensembles convexes de L-1 ou H-1. Sem. de Probabilites XIV. Lecture Notes in Mathematics 784, 220-222) in a general framework. More precisely, we say that the Kreps-Yan theorem is valid for a locally convex topological space (X, T), endowed with an order structure, if for each closed convex cone C in X such that C superset of X_ and C boolean AND X+ = {0}, there exists a strictly positive continuous linear functional on X, whose restriction to C is non-positive. We first show that the Kreps-Yan theorem is not valid for spaces L-p(ohm, F, P) if (ohm, F, P) fails to be sigma-finite. Then we prove that the Kreps-Yan theorem is valid for topological vector spaces in separating duality < X, Y >, provided Y satisfies both a "completeness condition" and a "Lindelof-like condition". We apply this result to the characterization of the no-arbitrage assumption in a general intertemporal framework. (c) 2004 Elsevier B.V. All rights reserved.