Consider the stochastic differential equation

$\displaystyle d S_t = \mu S_t dt + \sigma^2 S_t dW_t$ for all $\displaystyle t \in [0,T]$

where $\displaystyle \{ W_t \}t_0$ is a standard Brownian motion, $\displaystyle \mu$, $\displaystyle \sigma$ and $\displaystyle S_0$ are positive constants. Derive the unique solution to the above SDE.

Apparently this was derived in class but I'm missing that lecture so I guess I wasn't there or have lost it...

The actual solution is...

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

Anyone got any ideas on how to derive that?

First step is...

$\displaystyle S_t = s + \mu \int_0^T S_u du + \sigma \int_0^T S_u dW_u$

I seem to think it involves Ito process/lemmas...

Maybe I should do something like add in...

$\displaystyle \frac{\sigma^2}{2} - \frac{\sigma^2}{2}$