Geometric Brownian Motion

Definition

Consider a Brownian motion \fs4 B_t on a filtered space \fs4 (\Omega, \mathbb{F}, \mathcal{F}, P). A geometric Brownian motion (GBM) \fs4 X_t, with drift \fs4 \mu and volatility \fs4 \sigma is the solution of the stochastic differential equation

\fs4 dX_t = \mu X_t dt + \sigma X_t B_t

Features of the GBM

The geometric Brownian motion is routinely used as a model of stock prices, for example it underpins the Black-Scholes formula. One important feature is that the natural logarithm \fs4 x_t = X_t will follow a simple Brownian motion with drift. Applying the Ito formula to the above SDE for the function \fs4 x = \ln X yields

\fs4 dx_t = \left(\mu - \frac{1}{2}\sigma^2 \right)dt + \sigma dB_t

This implies that the GBM can be represented as

\fs4 X_t = X_0 \exp \left\{ \left( \mu - \frac{1}{2}\sigma^2 \right) t + \sigma B_t \right\}


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