Dominated families of martingale, supermartingale and quasimartingale laws

Consider a dominated family L of probability measures; we investigate the question of whether a single probability Q̂ ϵ L equivalent to the whole family L exists. We show that for supermartingale, quasimartingale and martingale laws the answer is positive. We then provide a necessary and sufficient condition for the existence of an equivalent (super, quasi) martingale measure and deduce an alternative characterization of semimartingales. We further study this problem in the context of security markets models and generalize the well-known fundamental theorem of asset pricing to cover the case of markets with frictions.