In the modern version of Arbitrage Pricing Theory suggested by Kabanov and Kramkov the fundamental financially meaningful concept is an asymptotic arbitrage. The "real world" large market is represented by a sequence of "models" and, though each of them is arbitrage free, investors may obtain non-risky profits in the limit. Mathematically, absence of the asymptotic arbitrage is expressed as contiguity of envelopes of the sets of equivalent martingale measures and objective probabilities. The classical theory deals with frictionless markets. In the present paper we extend it to markets with transaction costs. Assuming that each model admits consistent price systems, we relate them with families of probability measures and consider their upper and lower envelopes. The main result concerns the necessary and sufficient conditions for absence of asymptotic arbitrage opportunities of the first and second kinds expressed in terms of contiguity. We provide also more specific conditions involving Hellinger processes and give applications to particular models of large financial markets. * The authors warmly express their thanks to Yuri Kabanov for his helpful discussions and advices. † The first author acknowledges the hospitality of Lavinia Ostafe and Walter Schachermayer at the University of Vienna where this work was completed.

In the modern version of Arbitrage Pricing Theory suggested by Kabanov and Kramkov the fundamental financially meaningful concept is an asymptotic arbitrage. The "real world" large market is represented by a sequence of "models" and, though each of them is arbitrage free, investors may obtain non-risky profits in the limit. Mathematically, absence of the asymptotic arbitrage is expressed as contiguity of envelopes of the sets of equivalent martingale measures and objective probabilities. The classical theory deals with frictionless markets. In the present paper we extend it to markets with transaction costs. Assuming that each model admits consistent price systems, we relate them with families of probability measures and consider their upper and lower envelopes. The main result concerns the necessary and sufficient conditions for absence of asymptotic arbitrage opportunities of the first and second kinds expressed in terms of contiguity. We provide also more specific conditions involving Hellinger processes and give applications to particular models of large financial markets. * The authors warmly express their thanks to Yuri Kabanov for his helpful discussions and advices. † The first author acknowledges the hospitality of Lavinia Ostafe and Walter Schachermayer at the University of Vienna where this work was completed.

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