Martingale Measures for Discrete Time Processes with Infinite Horizon

Let (St)t2I be an IR {valued adapted stochastic process on ( ;F ; (Ft)t2I ; P ). A basic problem, occuring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on F equivalent to P such that (St)t2I is a martingale with respect to Q. It is known since the fundamental papers of Harrison{Kreps (79), Harrison{Pliska(81) and Kreps(81) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in nancial economics. We introduce the intermediate concept of "no free lunch with bounded risk". This is a somewhat more precise version of the notion of "no free lunch": It requires that there should be an absolute bound of the maximal loss occuring in the trading strategies considered in the de nition of "no free lunch". We shall give an argument why the condition of "no free lunch with bounded risk" should be satis ed by a reasonable model of the price process (St)t2I of a securities market. We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) in nite. A similar theorem was recently obtained by Delbaen (92) for the case of continuous time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous time case when the process (St)t2IR+ is bounded and | roughly speaking | the jumps occur at predictable times.