## Term Glossary

### 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: $$P_l(\delta X)=\lambda^{H} P_{\lambda t}(\lambda^{H} \delta X)$$

$P_l$
probability denisty (pdf) function of price returns
$\delta X$
return increment at scale $l$
$\lambda$
arbitrary number greater then zero
For white noise $H=0$, for brownian noise $H\approx 0.5$, while $H=1$ indicates directed motion.

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 $H=2-D$.

###### Function Reference
portfolio_hurstExponent, position_hurstExponent