A rough path approach to pathwise stochastic integration à la Föllmer

Abstract

We develop a general framework for pathwise stochastic integration that extends Föllmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of Itô, Stratonovich, and backward Itô integration. More precisely, for a continuous path admitting both quadratic variation and Lévy area along a fixed sequence of partitions, we define pathwise stochastic integrals as limits of general Riemann sums and prove that they coincide with integrals defined with respect to suitable rough paths. Furthermore, we identify necessary and sufficient conditions under which the quadratic variation and the Lévy area of a continuous path are invariant with respect to the choice of partition sequences.