Integral representations of some functionals of fractional Brownian motion

TL;DR

This paper develops generalized Itô formulas for functions of averages of fractional Brownian motion, valid for convex functions, providing integral representations in the generalized Lebesgue-Stieltjes sense.

Contribution

It introduces change of variables formulas for averages of fractional Brownian motion applicable to convex functions, extending classical Itô calculus.

Findings

Formulas valid for all convex functions

Integral representations in generalized Lebesgue-Stieltjes sense

Applicable to both arithmetic and geometric averages

Abstract

We prove change of variables formulas [It\^o formulas] for functions of both arithmetic and geometric averages of geometric fractional Brownian motion. They are valid for all convex functions, not only for smooth ones. These change of variables formulas provide us integral representations of functions of average in the sense of generalized Lebesgue-Stieltjes integral.