MARTINGALE MEASURES FOR DISCRETE‐TIME PROCESSES WITH INFINITE HORIZON

Abstract

Let (St)tεI be an Rd‐valued adapted stochastic process on (Ω, ?, (?t)tεI, P). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on ? equivalent to P such that (St)tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of “no arbitrage” and “no free lunch” in financial 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 an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of “no free lunch.” We give an argument as to why the condition of “no free lunch with bounded risk” should be satisfied by a reasonable model of the price process (St)tεI 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) infinite. A similar theorem was recently obtained by Delbaen (1992) for 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)tεR+ is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be “almost a martingale” in order to allow an equivalent martingale measure.

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