Equivalent supermartingale densities and measures in discrete time infinite horizon market models

Abstract

We consider a general discrete time infinite horizon securities market model in which the set $\ccW$ of stochastic wealth processes, corresponding to investment strategies, is subject to a number of axioms with financial interpretation. We obtain criteria for the existence of equivalent supermartingale densities and measures for the set $ \ccW _ + $ of nonnegative elements of $ \ccW $. These criteria are expressed in terms of various no-arbitrage conditions. The most complete results are formulated for Fatou closed sets $ \ccW $. This closedness condition is satisfied by the traditional market model with a finite number of basic assets. A feature of the paper consists of applying the Kreps–Yan theorem for the space $L ^\infty $ with the norm topology. With the use of this theorem we establish the existence of an equivalent supermartingale density under the absence of free lunch with vanishing risk condition for strategies with finite horizons.