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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?

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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.

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  • $\begingroup$ Why is mu=r-0.5(sigma)^2?Do they yield the same answers in a question? $\endgroup$
    – ankc
    Commented Dec 21, 2013 at 7:29
  • $\begingroup$ If both are mathematically the same shouldn't they have the same expectation and variance? $\endgroup$
    – ankc
    Commented Dec 21, 2013 at 7:56
  • $\begingroup$ They do have the same expectation and variance. The two references are using the same symbol to denote different things. $\endgroup$
    – Flounderer
    Commented Dec 21, 2013 at 20:57

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