### Hurst Exponent

Hurst exponent is a dimensionless estimator for the self-similarity of a time series.

Mathematically, a process \(X(t)\) is said to be self-similar of exponent \(H\), if for any \(\lambda\) > 0 the process is the same as \(\lambda^{-H}X(\lambda t)\). For a mono-fractal process being self-similar with exponent \(H\) implies that its probability denisty scales as follows: \begin{equation} P_l(\delta X)=\lambda^{H} P_{\lambda t}(\lambda^{H} \delta X) \end{equation}

- \(P_l\)
- probability denisty (pdf) function of price returns
- \(\delta X\)
- return increment at scale \(l\)
- \(\lambda\)
- arbitrary number greater then zero

Applied to financial data such as stock prices, the Hurst Exponent can be interpreted as a measure for the trendiness: \(H\lt0.5\) high volatility, stock price is anti-trended, \(H=0.5\), stock price behaves like a brownian process, no trend, \(H>0.5\) stock price has a trend. Some people believe that an estimation of the Hurst Exponent may yield some valuable information on the long-term behaviour of a particular stock.

For an affine processes like fractional Brownian Motion, the Hurst Exponent \(H\) is related to the fractal dimension \(D\) by the relation \(H=2-D\).