Random variables as pathwise integrals with respect to fractional Brownian motion

Yuliya Mishura, Georgiy Shevchenko, Esko Valkeila

Published 2011-11-08, updated 2012-09-20Version 2

We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.