Hi everyone,

modelling stock prices I have the following generalized Wiener process:

$\displaystyle dS_t=S_t(\mu d_t + \sigma dW_t)$

Applying Itô's lemma gives us:

$\displaystyle ln(S_t)-ln(S_0)=(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}$

with $\displaystyle \epsilon$ being a standard normal random variable with mean 0 and variance 1. Taking exp of the above, gives us:

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

That gives us the expected value of S_t which I don't understand.

$\displaystyle E[S_t]=S_0exp(\mu t})$

Specifically my question is: Why is the expected value of

$\displaystyle E[e^X]=e^{\mu +\frac{1}{2}\sigma^2} \qquad ?$

with X being a normal random variable with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$.

Thank you for explanation/proof, but also for relevant literature.