Existence of Lévy's area and pathwise integration

Abstract

Rough path analysis can be developed using the concept of con- trolled 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.