# Differences between different Geometric Brownian Motion

I'm currently studying brownian motion and came across two kinds of geometric brownian motion. http://homepage.univie.ac.at/kujtim.avdiu/dateien/BrownianMotion.pdf (page 14) http://www.math.unl.edu/~sdunbar1/MathematicalFinance/Lessons/StochasticCalculus/GeometricBrownianMotion/geometricbrownian.pdf (page 2)

I can't see the difference between the two formulas. I prefer the 2nd one since it seems simpler but both give different kind of expectation and variance and this is making it confusing for me to understand.

Can anyone explain the difference between the two?

Perhaps it's the use of $\mu$ in both formulas that is confusing you? The first reference gives the definition of geometric Brownian motion as $$\frac{dS_t}{S_t} = \mu dt + \sigma dW_t$$ and the second as $$\frac{dS_t}{S_t} = r dt + \sigma dW_t$$ so $\mu$ in the first reference is $r$ in the second. The second reference then uses $\mu$ as a notation for $r-\frac{1}{2}\sigma^2$. So in other words, the symbol $\mu$ in the second reference does not have the same meaning as the symbol $\mu$ in the first reference. But both are mathematically the same.