Towards a probability-free theory of continuous martingales

Abstract

Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with continuous price paths. This allows us to establish probability-free versions of a number of standard results in martingale theory, including the Dubins-Schwarz theorem, the Girsanov theorem, and results concerning the It\^o integral. We also establish the existence of an equity premium and a CAPM relationship in this probability-free setting.