On the fractional stochastic integration for random non-smooth integrands

TL;DR

This paper develops a new approach for stochastic integration of non-smooth, random integrands with respect to fractional Brownian motion for Hurst parameter H>1/2, extending the integral's applicability without requiring differentiability.

Contribution

It introduces a foresighted stochastic integral for non-smooth integrands with respect to fractional Brownian motion, avoiding Malliavin calculus and ensuring continuity at H=1/2.

Findings

Integral extends to a broad class of square integrable processes

Integration depends continuously on H near 1/2

No differentiability or Hölder conditions required for integrands

Abstract

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations; differentiability of H\"older property of any kind or degree is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring corresponding restrictions on the forecasting error. This integration is based on It\^o's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend continuously on H at H=1/2+0.