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Fractal Brownian Motion 📂Probability Theory

Fractal Brownian Motion

Definition

$E \left( X_{t} \right) = 0$ given $X_{t}$ is a Gaussian process and let’s denote it by $H \in (0, 1)$. The Fractional Brownian motion can be defined in the following two ways.

Definition through Covariance 1

The covariance at time point $t, s$ of $X_{t}$, if it follows the expression below, is referred to as Fractional Brownian Motion. $$ \operatorname{Cov} \left( X_{t}, X_{s} \right) = {{ 1 } \over { 2 }} \left( t^{2H} + s^{2H} - \left| t-s \right|^{2H} \right) $$

Definition through Conditions 2

$X_{t}$ is considered as Fractional Brownian Motion if it satisfies the following two conditions.

Explanation

The term Fractional is thought to be more appropriately derived from Fractal rather than Fraction considering the self-similarity mentioned in the definition, thus it is rationalized to Fractional Brownian Motion.

When $H = 1/2$, it is exactly a Brownian Motion. In other words, FBM is a perfect generalization of standard BM.


  1. Sottinen. (2003). Fractional Brownian Motion in Finance and Queueing: p7. ↩︎

  2. Yang. (2008). LRD of Fractional Brownian Motion and Application in Data Network: p6~8. ↩︎