Existence of L\'evy's area and pathwise integration

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.