On the closedness of sums of convex cones in \(L^0\) and the robust no-arbitrage property
This note is a natural complement to the previous paper of the authors [Finance Stoch. 6, No. 3, 371--382 (2002; Zbl 1026.60051)], where they studied no-arbitrage criteria for markets with efficient friction. They showed that for the model with efficient friction (i.e., non-empty interiors of the positive duals to the solvency cones) the existence of strictly consistent price system is equivalent to the strict no-arbitrage conditions for the whole interval \((NA^s)\). \textit{W. Schachermayer} [``The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time'' (Preprint, November 2001), Math. Finance 14, 19--48 (2004; Zbl 1119.91046)] proved that the existence of strictly consistent price system is equivalent to the newly introduced ``robust no-arbitrage property'' ensuring the closedness of the set of hedgeable claims. He provided an example showing that \((NA^s)\) does not guarantee this closedness and, hence, does not coincide with \((NA^r)\). In this article the authors give a synthesis and comparison of available results on no-arbitrage criteria in discrete time using advantages of more general geometric framework of their above mentioned paper to give quick proofs.
scientific article; zbMATH DE number 2169536