Random variables as pathwise integrals with respect to fractional Brownian motion

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.

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• Integral representation with respect to fractional Brownian motion under a log-Hölder assumption
• Replication of Wiener-transformable stochastic processes with application to financial markets with memory