Expectation Geometric Brownian Motion

Question

Can someone help show me a simple way to show:

$$\mathbb{E}(S_t)= S_0e^{\mu t}$$

for

$$ S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) $$

from this page: http://en.wikipedia.org/wiki/Geometric_Brownian_motion

Answer 1

Hint: For a normal random variable $X \sim N(\mu, \sigma),$ show that

$$\mathbb{E}[e^X]=e^{\mu+\frac 12 \sigma^2}$$