Rough Paths above Weierstrass Functions

Abstract

Rough paths theory allows for a pathwise theory of solutions to differential equations driven by highly irregular signals. The fundamental observation of rough paths theory is that if one can define "iterated integrals" above a signal, then one can construct solutions to differential equations driven by the signal. The typical examples of the signals of interest are stochastic processes such as (fractional) Brownian motion. However, rough paths theory is not inherently random and therefore can treat irregular deterministic driving signals such as a (multivariate) Weierstrass function. To the authors' best knowledge, no explicit construction of a rough path (the "iterated integrals") above a multivariate Weierstrass function has been constructed, nor has there been an explicit solution to a differential equation driven by a multivariate Weierstrass function. This note supplies a construction of a rough path above a multivariate Weierstrass function. We conclude with some illustrations and some examples of solving differential equations driven by rough Weierstrass functions.