The Derivatives-on-Line Pages

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The geometric Brownian motion

One very important special case comes when

$\displaystyle Y\left( t\right)$	$\displaystyle =$	$\displaystyle f\left[ X\left( t\right) \right]$ , where
$\displaystyle f\left( x\right)$	$\displaystyle =$	$\displaystyle \exp \left\{ x\right\}$ ,
$\displaystyle X\left( t\right)$	$\displaystyle =$	$\displaystyle \left( \mu -\frac{1}{2}\sigma ^{2}\right) dt+\sigma dB\left( t\right)$ .

The diffusion $Y\left( t\right)$ is a geometric Brownian motion, and its importance lies mainly in the fact that it is the simplest form of a diffusion that remains almost surely positive. Itô's formula would give in this case that

$\displaystyle dY\left( t\right)$	$\displaystyle =$	$\displaystyle \mu Y\left( t\right) dt+\sigma Y\left( t\right) dB\left( t\right)$ , or
$\displaystyle \frac{dY\left( t\right) }{Y\left( t\right) }$	$\displaystyle =$	$\displaystyle \mu dt+\sigma dB\left( t\right)$ .

In figure 7.1 one can find a sample path for a geometric Brownian motion, with $\mu =1$ and $\sigma =2$ . One can observe that such a pattern could well match the pattern of a financial asset.

Kyriakos 2003-03-17