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

Maximum of Wiener Brownian Motion

Let W_0 be the standard Wiener Brownian Motion process, and M(t) defined as M(t)=\max_{0 \leq s \leq t} W_{0}(t), we then have the following result:

P(M(t) \geq x)= 2(1-\phi(\frac{x}{\sqrt{t}}))

where \phi denotes the standard normal cumulative distribution function:

\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^{2}/2} \forall{z}\in\mathbb{R}

This result is simply a direct application of Theorem 2 from this paper with \alpha=0

May 4, 2009 Posted by | Wiener Brownian Motion | Leave a comment