Fractals and Stochastic Calculus

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Standard Deviation of Fractional Brownian Motion

Let B_{H}(t) be the zero-mean Fractional Brownian Motion process of Hurst parameter H (0 \le H \le 1), such a process is defined as having the following covariance structure:

E[B_{H}(t)B_{H}(s)]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}) (1)

Then(by taking s=t in (1))

Var(B_{H}(t))=E[(B_{H}(t)-E(B_{H}(t)))^2]=E[B_{H}^{2}(t)]=|t|^{2H}

Which gives the standard deviation as |t|^H (and for the Wiener Brownian Motion, we indeed get a standard deviation of \sqrt{t})

May 5, 2009 Posted by | Fractional Brownian Motion | Leave a comment