# Geometric Brownian Motion

Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$

$$dS = \mu Sdt + \sigma SdX$$

Show by the application of Itô's Lemma to function $\log S$ that

$$d \log S = \left ( \mu - \frac12 \sigma^2 \right ) dt + \sigma dX$$

Then by integrating $d \log \it S$ from $0$ to $t$ and taking expectations show the expected value of $S_t$ condition by the initial value $S_0$ is

$$\Bbb E \left [ S_t\mid S_0 \right ] = S_0e^{ut}.$$

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Right, so at the moment what we have is the verbatim transcript of your homework. Now, what are your thoughts about it? –  Did Mar 5 '13 at 10:08
What is $u$ here? –  Ilya Mar 5 '13 at 10:11
$\mu$ is the drift term in the stochastic process. Plus I have just discovered mathjax and am very impressed with it. Easy to pick up. –  Barzillai Mar 5 '13 at 10:40
You will find a solution to your question in this post math.stackexchange.com/questions/126870/… –  Zbigniew May 24 '13 at 7:22