Robust No Arbitrage Condition for Continuous-Time Models with Transaction Costs

In frictionless markets, the absence of arbitrage opportunities is equivalent to the existence of a martingale process evolving in the ray R_ S where S is the d-dimensional price process (whose first component is the numeraire). With transaction costs, absence of arbitrage opportunities is related to the existence of a consistent price system; It plays the same role as the martingale measure in the frictionless case. This is a martingale evolving in the solvency cone. The solvency cone is the set of all portfolios one can change, paying transaction costs, into another ones which the position in every asset is nonnegative. The Robust No Free Lunch condition RNFL means that the absence of asymptotic arbitrage opportunities condition still holds in presence of transaction costs smaller than those of the initial model. We extend the RNFL theorem formulated for discrete-time models to a general continuous-time setting; We prove that Condition RNFL is still equivalent to the existence of a strictly consistent price system, i.e. a consistent price system evolving in the interior of the solvency cone. The existence of such processes is assumed in the formulation of super–hedging prices characterization of European and American payo s. We also provide a characterization of absence of arbitrage for abstract continuous-time models we relate to the existence of a strict consistent price system