# Fractals and Stochastic Calculus

## 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

## 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