Asymptotic arbitrage in large financial markets

Abstract.

A large financial market is described by a sequence of standard general models of continuous trading. It turns out that the absence of asymptotic arbitrage of the first kind is equivalent to the contiguity of sequence of objective probabilities with respect to the sequence of upper envelopes of equivalent martingale measures, while absence of asymptotic arbitrage of the second kind is equivalent to the contiguity of the sequence of lower envelopes of equivalent martingale measures with respect to the sequence of objective probabilities. We express criteria of contiguity in terms of the Hellinger processes. As examples, we study a large market with asset prices given by linear stochastic equations which may have random volatilities, the Ross Arbitrage Pricing Model, and a discrete-time model with two assets and infinite horizon. The suggested theory can be considered as a natural extension of Arbirage Pricing Theory covering the continuous as well as the discrete time case.