Fractals and Stochastic Calculus

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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

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May 4, 2009 - Posted by | Wiener Brownian Motion

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