Ito calculus without probability in idealized financial markets

TL;DR

This paper establishes the existence of quadratic variation for typical price paths in idealized financial markets without relying on probability, enabling pathwise Ito calculus applications.

Contribution

It proves quadratic variation exists for typical price paths in a probability-free setting, broadening the scope of pathwise Ito calculus in financial modeling.

Findings

Quadratic variation exists for typical cadlag price paths.

Pathwise Ito calculus can be applied without probabilistic assumptions.

Results enable new analysis methods in idealized markets.

Abstract

We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks only one monetary unit and brings infinite capital if quadratic variation does not exist. This result allows one to apply numerous known results in pathwise Ito calculus to typical price paths; we give a brief overview of such results.