BDG inequalities and their applications for model-free continuous price paths with instant enforcement

TL;DR

This paper develops an outer measure framework for model-free continuous price paths, establishing BDG inequalities and an Itô-type integral under instant enforcement, advancing the theory of pathwise stochastic calculus.

Contribution

It introduces an outer measure for instantly blockable properties and proves BDG inequalities and Itô's isometry in a model-free setting.

Findings

Established an outer measure assigning zero to instantly blockable sets.

Proved BDG inequalities and Itô's isometry for the new measure.

Analyzed properties of quadratic variation for model-free continuous martingales.

Abstract

Shafer and Vovk introduce in their book \cite{ShaferVovk:2018} the notion of \emph{instant enforcement} and \emph{instantly blockable} properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of ''typical'' price paths. In this paper we introduce an outer measure on the space $[0, +\ns) \times \Omega$ which assigns zero value exactly to those sets (properties) of pairs of time $t$ and an elementary event $\omega$ which are instantly blockable. Next, for a slightly modified measure, we prove It\^o's isometry and BDG inequalities, and then use them to define an It\^o-type integral. Additionally, we prove few properties for the quadratic variation of model-free, continuous martingales, which hold with instant enforcement.