Ah OK, its a little clearer.
But how can I expand the Doleans exponential? to get the fourth moment?
This just seems to be the same as applying ito to log X(t)?
If that is ocrrect then I follow you to this point, but how then do I build on this to get the 4th moment?
Just expand exp with the usual series exp(x)=1+x+x^2/2+x^3/6+x^4/24+...
Of course, to do things properly, you would want to scale time by c, expand, switch order of expectation and sum, and collect the terms with suitable powers of c (because the martingale property tells you
Correct me If I am wrong but in the above expansion each term divided by the relevant factorial correspends to the given moment. So the fourth moment
. However you still have to calculate the expectaionof x^4 so I think you wil end up using Ito's lemma anyway right?
With this in mind:
is equal to the variance t/n
This appears to be in the right direction but is still wrong.
Can you clarify my first point as well as where I am going wrong in my application of ito's lemma.