Rough paths and regularization

TL;DR

This paper explores the connection between rough paths theory and regularization methods in stochastic calculus, focusing on weak Dirichlet processes and their integrals, bridging deterministic and probabilistic approaches.

Contribution

It establishes a link between stochastically controlled processes and weak Dirichlet processes, and relates rough and Stratonovich integrals for specific classes of stochastic processes.

Findings

Connected rough paths and regularization techniques in stochastic calculus.

Linked weak Dirichlet processes with rough paths theory.

Compared rough and Stratonovich integrals for certain stochastic processes.

Abstract

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