The Existence of Absolutely Continuous Local Martingale Measures

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schachermayer (1994a) we showed that for locally bounded finite dimensional stochastic price processes S, the existence of an equivalent (local) martingale measure – sometimes called risk neutral measure – is equivalent to a property called No Free Lunch with Vanishing Risk (NFLVR). We also proved that if the set of (local) martingale measures contains more than one element, then necessarily, there are non equivalent absolutely continuous local martingale measures for the process S. We also gave an example, see Delbaen and Schachermayer (1994a) Example 7.7, of a process that does not admit an equivalent (local) martingale measure but for which there is a martingale measure that is absolutely continuous. The example moreover satisfies the weaker property of No Arbitrage with respect to general admissible integrands. We were therefore lead to the investigation of the relation between the two properties, the existence of an absolutely continuous martingale measure (ACMM) and the absence of arbitrage for general admissible integrands (NA). From an economic viewpoint a local martingale measure Q, that gives zero measure to a non negligible event, say F , poses some problems. The price of the contingent claim that pays one 1991 Mathematics Subject Classification. 90A09,60G44, 46N10,47N10,60H05,60G40.