Covariance of stochastic integrals with respect to fractional Brownian motion

TL;DR

This paper derives an explicit formula for the cross-covariance of stochastic integrals with respect to fractional Brownian motion, revealing properties of these integrals and providing alternative proofs for known results.

Contribution

It presents a new explicit expression for the cross-covariance of stochastic integrals with respect to fractional Brownian motion, offering insights into their structure and properties.

Findings

Derived explicit cross-covariance formula for stochastic integrals

Provided an alternative proof regarding fractional Bessel process

Showed the integral component is not a fractional Brownian motion

Abstract

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart's result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.