  Rough path analysis can be developed using the concept of controlled paths,
and with respect to a topology in which L\'evy's area plays a role. For vectors
of irregular paths we investigate the relationship between the property of
being controlled and the existence of associated L\'evy areas. For two paths,
one of which is controlled by the other, a pathwise construction of the L\'evy
area and therefore of mutual stochastic integrals is possible. If the existence
of quadratic variation along a sequence of partitions is guaranteed, this leads
us to a study of the pathwise change of variable (It\^o) formula in the spirit
of F\"ollmer, from the perspective of controlled paths.
