Equivalent martingale measures and no-arbitrage in stochastic securities market models

Abstract

We characterize those vector-valued stochastic processes (with a finite index set and defined on an arbitrary stochasic base) which can become a martingale under an equivalent change of measure.This question is important in a widely studied problem which arises in the theory of finite period securities markets with one riskless bond and a finite number of risky stocks. In this setting, our characterization gives a criterion for recognizing when a securities market model allows for no arbitrage opportunities (free lunches). Intuitively, this can be interpreted as saying if one cannot win betting on a process, then it must be a martingale under an equivalent measure, and provides a converse to the classical notion that one cannot win betting on a martingale.