Random variables as pathwise integrals with respect to fractional Brownian motion

TL;DR

This paper demonstrates that pathwise integrals with respect to fractional Brownian motion can represent any distribution, providing conditions for such representations and exploring applications in financial modeling.

Contribution

It establishes necessary and sufficient conditions for representing random variables as pathwise integrals with fractional Brownian motion, and shows all random variables can be approximated in an improper sense.

Findings

Any prescribed distribution can be achieved by such integrals.

Necessary and sufficient conditions for representation are provided.

Applications to fractional Black--Scholes model are discussed.

Abstract

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.