Financial Applications of Fractional Brownian Motion

Long-range Dependence in Economics and Finance

As mentioned in the paper (WTT99), long-range dependence in economics and finance has a long history and is an area of active research (e.g., see (Lo91), (CKW95)). The importance of long-range dependent processes as stochastic models lies in the fact that they provide an explanation and interpretation of an empirical law that is commonly referred to as the Hurst law or Hurst effect. In short, for a given set of observations $\left\{ {X_{i,} i \ge 1} \right\}$ with partial sum $Y\left( n \right) = \sum\limits_{i = 1}^n {X_i ,n \ge 1,}$ and sample variance $S^2 \left( n \right) = n^{ - 1} \sum\limits_{i = 1}^n {\left( {X_i - n^{ - 1} Y\left( n \right)} \right)^2 ,n \ge 1,}$ the rescaled adjusted range statistic or R/S-statistic is defined by

$$\frac{R}{S}\left( n \right) = \frac{1}{{S\left( n \right)}}\left( {\mathop {\max }\limits_{0 \le t \le n} \left( {Y\left( t \right) - \frac{t}{n}Y\left( n \right)} \right) - \mathop {\min }\limits_{0 \le t \le n} \left( {Y\left( t \right) - \frac{t}{n}Y\left( n \right)} \right)} \right),n \ge 1.$$

Hurst in (Hur51) found that many naturally occurring empirical records appear to be well represented by the relation $E\left( {\left( {R/S} \right)\left( n \right)} \right) \sim c_1 n^H$ as $n \to \infty$ with typical values of the Hurst parameter $H \in \left( {1/2,1} \right)$, and c ₁ a finite positive constant not depending on n. But in the case when the observations come from a short-range dependent model, then $E\left( {R/S\left( n \right)} \right) \sim c_2 n^{1/2}$ as $n \to \infty$, where c ₂ does not depend on n. The discrepancy between these two relations is called the Hurst effect or Hurst phenomenon. The analysis of the R/S-statistic, provided in (WTT99), (TTW95) and (TT97), leads to the recommendation to use a diverse portfolio of time-domain-based and frequency-domain-based graphics and statistical methods, including the graphical R/S-method, the modified R/S-statistic (Lo91) and Whittle’s approach. Also, another (possibly, surprising) recommendation is: in the case when statistical analysis cannot be expected to provide a definitive answer concerning the presence or absence of long-range dependence in asset price returns, a more revealing and also much more challenging approach to tackle this problem consists of providing a mathematically rigorous physical “explanation” for the presence or absence of the long-range dependence phenomenon in stock returns.