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A stochastic process S_{t} is said to follow a GBM if it satisfies the following stochastic differential equation[?]:
where {W_{t}} is a Wiener process or Brownian motion and u ('the percentage drift') and v ('the percentage volatility') are constants.
The equation has a analytic solution:
for an arbitrary initial value S_{0}. The correctness of the solution can be verified using Ito's Lemma. The random variable log( S_{t}/S_{0}) is Normally distributed with mean (uv.v/2).t and variance (v.v).t, which reflects the fact that increments of a GBM are Normal relative to the current price, which is why the process has the name 'geometric'.
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