Towards a probability-free theory of continuous martingales

TL;DR

This paper develops a probability-free framework for continuous martingales in financial markets, establishing key theorems and relationships without relying on probability theory.

Contribution

It introduces a novel probability-free approach to martingale theory, including versions of classical theorems and financial models.

Findings

Probability-free versions of Dubins-Schwarz and Girsanov theorems

Existence of equity premium without probability assumptions

CAPM relationship derived in a probability-free context

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.