Trajectory Based Models, Arbitrage and Continuity

Abstract: The paper develops no arbitrage results for trajectory based models by imposing general constraints on the trading portfolios. The main condition imposed, in order to avoid arbitrage opportunities, is a local continuity requirement on the final portfolio value considered as a functional on the trajectory space. The paper shows this to be a natural requirement by proving that a large class of practical trading strategies, defined by means of trajectory based stopping times, give rise to locally continuous functionals. The theory is illustrated, with some detail, for two specific trajectory models of practical interest. The implications for stochastic models which are not semimartingales are described. The present paper extends some of the results in [1] by incorporating in the formalism a larger set of trading portfolios.