Abstract:

We study a notion of pathwise integral, defined as the limit of non-anticipative Riemann sums, with respect to paths of finite quadratic variation, extending an idea of  H. Follmer (1979) to path-dependent integrands. We show that this integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals.
This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands and obtain a pathwise `signal plus noise' decomposition, which is a deterministic analog of the semimartingale decomposition, for a large class of irregular paths obtained through functional transformations of a reference path with non-vanishing quadratic variation. The relation with controlled rough paths is discussed.
Joint work with Anna ANANOVA (Imperial College London)