We provide a brief account of some aspects of the Arbitrage Theory in its historical development. The aim of this talk (intended to be accessible to graduate students in probability) is to explain the logic leading to a study of various types of stochastic deflators.
The supermartingale (local martingale) deflator is a strictly positive multiplier transforming value processes of admissible portfolios, i.e. dynamically rebalanced baskets of assets traded on the market, into supermartingales (local martingales). It may happen that the reciprocal of a deflator is the value process of a "market" portfolio. A "traded" supermartingale deflator is always unique.
The Takaoka theorem claims that in the very general model of financial market, with a finite-dimensional semimartingale price process describing the dynamics of basic securities, the absence of asymptotic arbitrage (the so-called the NAA1-property) is equivalent to the existence of a local martingale deflator. One can claim more (K.-Kardaras): the NAA1-property hold if and only if in any neighborhood of the "objective" probability there is an equivalent one with respect to which there exists the traded local martingale deflator.