Rough Paths and Regularization

Calculus via regularizations and rough paths are two methods to approach stochastic integration calculus close pathwise calculus. The origin of theory is purely deterministic, regularization based on deterministic techniques but there still a probability in the background. goal this paper establish connection between stochastically controlled-type processes, concept reminiscent from theory, so-called weak Dirichlet processes. As by-product, we present Stratonovich integrals for cadlag processes integrands continuous semimartingales integrators.