Fractional Brownian Motion

One dial H turns Brownian motion's independent increments into memory: persistent above 12, antipersistent below.

The idea

Fractional Brownian motion extends Brownian motion to quantities whose increments are correlated. Brownian motion assumes no memory: what the process does next is independent of everything it has already done. Many records violate that assumption — river levels flood in clustered decades, network traffic arrives in bursts, market volatility comes in calm and stormy spells — and in each, a large move makes the next large move more likely.

Fractional Brownian motion keeps the other features of Brownian motion — Gaussian values, an increment law depending only on elapsed time, self-similarity — and gives up independence alone. A single parameter $H$ sets the strength and direction of the correlation.

Definition (Fractional Brownian motion).

Fractional Brownian motion $B^{H}$ with Hurst parameter $H$ in $(0, 1)$ is the centered Gaussian process — every finite collection of its values is jointly normal with mean $0$ — that starts at $B^{H}_{0} = 0$ and has covariance $\mathbb{E}\big[B^{H}_{t}\,B^{H}_{s}\big] = \tfrac{1}{2}\big(t^{2H} + s^{2H} - |t-s|^{2H}\big).$

Because the process is centered and Gaussian, that one function is its whole law: any two centered Gaussian processes with the same covariance are the same process in distribution.

Since a centered Gaussian process is determined by its covariance, this one line carries every feature of the model. Setting $s = t$ gives the variance at a single time. Setting $H = \tfrac{1}{2}$ recovers Brownian motion, so the old model is a member of the new family. Applied to two adjacent increments, the covariance produces a number whose sign says whether the path tends to continue in its current direction, to reverse it, or neither.

Ways to work on it

• Walkthrough. Define the process by its covariance and see how the Hurst parameter shapes its memory and roughness.
• Proof. Prove self-similarity by rescaling the covariance, using that a centered Gaussian process is determined by its covariance.
• Practice. Compute variances and scaling factors, and classify the process's behavior from its Hurst parameter.
• Hardest. Quantify the long-range dependence of increments, or recover the Hurst parameter from observed variances.

Not sure where to start? Take the ten-question placement test.