A C\`adl\`ag Rough Path Foundation for Robust Finance

TL;DR

This paper develops a pathwise rough path framework for financial modeling, accommodating various trading strategies and market uncertainties, and introduces the Property (RIE) for cadlag paths to ensure robust integration.

Contribution

It introduces the Property (RIE) for cadlag paths, enabling a pathwise rough path foundation for stochastic integration in finance, including non-gradient strategies and universal portfolios.

Findings

Property (RIE) implies existence of cadlag rough paths and quadratic variation.

Pathwise rough integrals are limits of Riemann sums along suitable partitions.

Functional trading strategies and universal portfolios satisfy Property (RIE).

Abstract

Using rough path theory, we provide a pathwise foundation for stochastic It\^o integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for c\`adl\`ag paths, which is shown to imply the existence of a c\`adl\`ag rough path and of quadratic variation in the sense of F\"ollmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover's universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.