  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.
