A Question About Covariation Between Fractional Brownian Motions

Summary

This post asks how to calculate the covariance between increments of two fractional Brownian motions with different Hurst parameters. It proposes using the Riemann–Liouville representation, expressing each process through an integral against the same standard Brownian motion, and derives a candidate time-dependent covariance for the increments.

The post checks the expression against the ordinary Brownian case, where both Hurst parameters equal one half, and asks whether the derivation is correct. It provides no answer or independent validation, so the proposed formula should be treated as an unresolved conjecture rather than an established result. The setup also leaves the meaning of the conditional increment notation and assumptions about how the two processes are coupled unspecified; those details matter when defining cross-covariance.

Key ideas

• The post considers cross-covariance between fractional Brownian motions with unequal Hurst parameters.
• It proposes deriving the covariance from Riemann–Liouville representations driven by a common Brownian motion.
• The proposed expression is checked against the standard Brownian special case.
• No answer is supplied, and the coupling and increment assumptions remain unspecified.

Tags

# Covariance of (fractional) Brownian motions with different Hurst parameters


# Covariance of (fractional) Brownian motions with different Hurst parameters












I'd like to calculate the covariance function for fractional Brownian motions $$ E_t \left[ dW^H(t) dW^{H'}(t) \right] $$ but where the Hurst parameters are not equal: $H \neq H'$.

My first idea would be to look at the Riemann-Liouville representation of fractional Brownian motion (see for example Wikipedia Fractional Brownian Motion) and proceed from there.

Is there a better/more correct way?

EDIT:

According to the RLfBM representation: $$ W^H(t) = \frac{1}{\Gamma (H+ 1/2)} \int_0^t (t-s)^{H-1/2} \, dW(s) $$ with $dW(s)$ denoting the standard Brownian motion (i.e. $H=1/2$).

So I think that $$ E_t \left[ dW^H(t) dW^{H'}(t) \right] = \frac{t^{H + H' - 1}}{\Gamma (H+ 1/2)\Gamma (H' + 1/2)} dt $$ In particular this gives $E_t \left[ dW^H(t) dW^{H'}(t) \right] = dt$ when $H=H' = 1/2$. Is this right?

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