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Geometric Brownian motion
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== Arithmetic Brownian Motion == The process for <math>X_t = \ln \frac{S_t}{S_0}</math>, satisfying the SDE : <math>d X_t = \left(\mu -\frac{\sigma^2}{2}\,\right) dt + \sigma dW_t\, ,</math> or more generally the process solving the SDE : <math>d X_t = m\, dt + v\, dW_t\, ,</math> where <math>m</math> and <math>v >0</math> are real constants and for an initial condition <math>X_0</math>, is called an Arithmetic Brownian Motion (ABM). This was the model postulated by [[Louis Bachelier]] in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as [[Bachelier model]]. As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.
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