1. ## Stochastic differential equation

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...

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

2. Ok I found another answer elsewhere but I still have questions...

http://mplab.ucsd.edu/tutorials/sde.pdf (end of pg 13-14)

Here is the solution I found except I replace $\displaystyle X_t$ with $\displaystyle S_t$ and $\displaystyle B_t$ with $\displaystyle W_t$

Using Ito's rule on $\displaystyle \log(S_t)$ we get (Why $\displaystyle \log(S_t)$???).
$\displaystyle d \log S_t = \frac{1}{S_t}dS_t + \frac{1}{2} \left ( -\frac{1}{S_t^2} \right )^2 (dS_t)^2$

$\displaystyle = \frac{dS_t}{S_t} - \frac{1}{2} \sigma^2 dt$

$\displaystyle = \left ( \mu S_t - \frac{1}{2} \sigma^2 \right )dt + \alpha dW_t$

Thus, And thus solution...
$\displaystyle \log S_t = \log S_0 + \left ( \mu - \frac{1}{2}\sigma^2 \right )t + \sigma W_t$

And thus solution...

I honestly can't justify any of those steps.

3. Ok perhaps I can justify the first step.

Ito lemma (process?)

$\displaystyle dF(x_t) = F'(X_t)g_tdt + F'(X_t)h_tdW_t + \frac{1}{2}F''(X_t)d_t^2 dt$ as written in my notes.

So!

$\displaystyle d \log S_t = \mu \frac{1}{S_t} dS_t + \sigma \frac{1}{S_t} dW_t + \frac{1}{2}\left (-\frac{1}{S_t^2} \right )^2 (dS_t)^2$

But why does the middle term disappear (I guess it is = to 0 somehow that I'm not seeing)
And what happens to the $\displaystyle \mu$ and $\displaystyle \sigma$ terms? Why were they left out at this step in the solution I found.