Integral representation with respect to fractional Brownian motion under a log-H\"{o}lder assumption

TL;DR

This paper proves that certain log-Hölder continuous adapted processes can be represented as stochastic integrals with respect to fractional Brownian motion, extending the fractional integral definition.

Contribution

It introduces a new integral representation for adapted processes under log-Hölder continuity, broadening the class of processes representable via fractional Brownian motion.

Findings

Representation of final values as fractional Brownian motion integrals

Extension of fractional integral definition

Applicable to log-Hölder continuous processes

Abstract

We show that if a random variable is the final value of an adapted log-H\"{o}lder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with adapted integrand. In order to establish this representation result, we extend the definition of the fractional integral.