Wiki Wiki Web

Hurst Exponent

A self-affine random process $X(t)$ satisifes: $$X(ct) = c^H X(t)$ For $H=1/2,t=1$, we get a $\sqrt{t}$ scaling rule typical of random walks.

Looking at the fractional bronwian motion $B_h(t)$ with: $$E(B_H(t+T)-B_H(t)) = 0$$ $$E\left((B_H(t+T)-B_H(t))^2\right) = T^{2H}$$

The correlation of $\frac{1}{\tau} (B_H(t+\tau)-B_H(t))$ with $\frac{1}{\tau} (B_H(t)-B_H(t-\tau))$ is $2^{2H-1}-1$. This means that the FBM is either trending or mean-reverting, and this correlation does not depend on $\tau$.

Estimation of the Hurst Exponent

$$ S_k = \sum_{i=1}^k (x_i - m_N), 1 \leq k \leq N $$ $$ R/S = \frac{1}{\sigma_N} (\max(S_k)-\min(S_k)), 1 \leq k \leq N $$ The hurst exponent can be estimated via $$ R/S = (aN)^H $$ by plotting $\ln R/S$ against $\ln N$, the Hurst exponent is the slope.