Geometric rough paths on infinite dimensional spaces

TL;DR

This paper extends the theory of rough paths to infinite-dimensional Banach and Hilbert spaces, providing approximation criteria and applications to stochastic analysis.

Contribution

It introduces criteria for approximating Banach space-valued rough paths by bounded variation signatures and applies these to Hilbert spaces and martingale analysis.

Findings

Criteria established for approximation of rough paths in Banach spaces.

Application of results to Hilbert space rough paths.

Wong-Zakai type approximation results for function space martingales.

Abstract

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $\alpha$-rough paths by signatures of curves of bounded variation, given some tuning of the H\"older parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.