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I haven’t done much work in mathematics lately but I spent a lot of time working with fractional Brownian motion (fBm) when doing my PhD and when I was first learning to code. Fractional Brownian motion is a process which generalizes ordinary Brownian motion, a stochastic process with stationary independent Gaussian increments.

Fractional Brownian motion generalizes Brownian motion in such a way that its increments are no longer necessarily independent. The process \( (B^H_t, t \geq 0) \) differs from Brownian motion in that its covariance function is

$$ \mathbb{E}\left[B^H_t B^H_s\right] = \frac{1}{2}\left( s^{2H} + t^{2H} - |t-s|^{2H} \right) $$

for \( 0 < t < s \), where \( H \in (0, 1) \) is known as the Hurst parameter. When \( H = \frac{1}{2} \), the process is a Brownian motion with independent increments. If \( H \in (\frac{1}{2}, 1) \), then the increments of the process are positively correlated. If \( H \in (0, \frac{1}{2}) \), then the increments of the process are negatively correlated.

A well-studied problem related to fBm is the estimation of the Hurst parameter from discrete samples of an fBm realization. One of the best estimators of \( H \) uses the quadratic variation (or 2-variation) of samples \( \{B^H_{t_k}\}_{k=0}^n \)

$$ V_n = \frac{1}{n}\sum_{k=1}^n \left( B_{t_k}^H - B_{t_{k-1}}^H \right)^2 $$

which converges to \( (\Delta t)^{2H} \) as \( n \to \infty \) where \( \Delta t = t_k - t_{k-1} \). This is a consequence of ergodic theory, specifically the Birkhoff-Khinchin theorem. The result leads to the estimator

$$ \widehat{H} = \frac{\log (V_n)}{2 \log (\Delta t)} $$

This works well for *standard* fBm, but is not useful when the fBm process is scaled with a coefficient. For scaled fBm we can use a direction-change estimator, which I stumbled upon as a graduate student writing my dissertation (it’s included as an appendix). Here is an abridged derivation.

Consider a discretely observed fBm process \( \{B^H_{t_k}\}_{k=0}^n \) and let \( Z_k = B^H_{t_{k+1}} - B^H_{t_k} \) be the fractional Gaussian noise (fGn) sequence for \( k = 0, \ldots, n-1 \). Let \( p \) be the proportion of fGn increments which change direction from the previous increment:

$$ p = \frac{1}{n-1} \sum_{k=1}^{n-1} \mathbf{1}_{{\text{sgn}(Z_k) \neq \text{sgn}(Z_{k-1})}}(k) $$

It has expected value

$$ \mathbb{E}[p] = \frac{1}{n-1} \sum_{k=1}^{n-1} \mathbb{P}\left( \text{sgn}(Z_k) \neq \text{sgn}(Z_{k-1}) \right) $$

By symmetry, the probability terms are

$$ \mathbb{P}\left( \text{sgn}(Z_k) \neq \text{sgn}(Z_{k-1}) \right) = \mathbb{P}\left( Z_k < 0 \mid Z_{k-1} > 0 \right) $$

The autocovariance of fGn is given by

$$ \gamma_H(k) = \frac{1}{2} \left[ |k-1|^{2H} - 2k^{2H} + |k+1|^{2H} \right] $$

For \( k \) and \( k+1 \) corresponding to the respective times \( t = k \) and \( t = k+1 \), the unconditioned distribution of any \( Z_k = B^H_{t_{k+1}} - B^H_{t_k} \sim \mathcal{N}(0,1) \). Thus, the multivariate distribution of \( Z_k \) and \( Z_{k-1} \) is

$$ \left( \begin{array}{c} Z_k \\ Z_{k-1} \end{array} \right) \sim \mathcal{N}\left[\left(\begin{array}{c} 0 \\ 0 \end{array}\right) ,\left( \begin{array}{cc} \gamma_H(0) & \gamma_H(1) \\ \gamma_H(1) & \gamma_H(0) \end{array} \right)\right] $$

where \( \gamma_H(0) = 1 \) and \( \gamma_H(1) = 2^{2H-1} - 1 \).

Then it follows that the conditional distribution is

$$ Z_k \mid Z_{k-1} \sim \mathcal{N}\left(\gamma_H(1) Z_{k-1}, 1-\gamma_H(1)^2\right) $$

Based on this result, the direction change probabilities can be written by the integral

$$ \mathbb{P}\left( Z_k < 0 \mid Z_{k-1} > 0 \right) = \frac{1}{2\pi} \int_0^\infty \int_{-\infty}^{-\frac{\gamma_H(1) x}{\sqrt{1-\gamma_H(1)^2}}} \exp\left( -\frac{y^2+x^2}{2} \right) \,\text{d} y \,\text{d} x $$

Integration using polar coordinate transformation gives

$$ \mathbb{P}\left( Z_k < 0 \mid Z_{k-1} > 0 \right) = \frac{1}{2} + \frac{1}{\pi} \arctan \left( - \frac{\gamma_H(1)}{\sqrt{1-\gamma_H(1)^2}} \right) $$

This leads to the result (after summation) of

$$ \mathbb{E}[p] = \frac{1}{2} + \frac{1}{\pi} \arctan \left( - \frac{\gamma_H(1)}{\sqrt{1-\gamma_H(1)^2}} \right) $$

If we estimate \( \mathbb{E}[p] \) with \( p \) and solve for \( \gamma_H(1) \) we get

$$ \gamma_H(1) = \frac{-A_p}{\sqrt{1 + (-A_p)^2}} $$

where

$$ A_p = \tan\left( (p-\frac{1}{2})\pi \right) $$

Solving for \( H \) gives us the estimator

$$ \widehat{H}^d = \frac{1}{2} \left[ \frac{\log \left( -\frac{A_p}{\sqrt{1 + (-A_p)^2}} + 1 \right)}{\log 2} + 1 \right] $$

which we bound by the interval \( [0, 1] \). This estimator depends **only** on \( p \) (the proportion of increments of the fBm realization where a direction change occurs from the previous increment)!

This result also provides some interesting insight on the behavior of fBm direction changes as \( H \) approaches 0 and 1.

As \( H \to 0 \), the value of \( \gamma_H(1) \to -\frac{1}{2} \). Therefore,

$$ \lim_{H \to 0} \mathbb{E}[p] = \frac{2}{3} $$

Similarly, for \( H \to 1 \), the value of \( \gamma_H(1) \to 1 \), so

$$ \lim_{H \to 1} \mathbb{E}[p] = 0 $$

- fractional Brownian motion
- Brownian motion
- stationarity
- independence
- Hurst index
- quadratic variation
- Birkhoff-Khinchin theorem