On low dimensional case in the fundamental asset pricing theorem with transaction costs

The well-known Harrison-Plisse theorem claims that in the classical discrete time model of the financial market with finite Omega there is no arbitrage iff there exists an equivalent martingale measure. The famous Dalang-Morton-Willinger theorem extends this result for an arbitrary Omega. Kabanov and Stricker [KS01] generalized the Harrison-Pliska theorem for the case of the market with proportional transaction costs. Nevertheless the corresponding extension of the Kabanov and Stricker result to the case of non-finite Omega fails, the corresponding counter-example with 4 assets was constructed by Schachermayer [S04]. The main result of this paper is that in the special case of 2 assets the Kabanov and Stricker theorem can be extended for an arbitrary Omega. This is quite a surprising result since the corresponding cone of hedgeable claims (A) over capT is not necessarily closed.