This is rather an unfamiliar territory for me, but I managed to do the first part of the question, so there is no reason why I cannot do the second, especially with some help ))) Besides, I really want to understand how I can use MGF here.

Question

If and , ie ,

where is standard Brownian motion, show that

.

Show also that . (Hint: consider the moment generating function of the standard normal variable .

Answer.

I use Ito's lemma to prove the first:

first and second derivatives of

(as the first bracket is zero.)

Now, for the second part, I could try substitution the formula for w_t :

- somehow I feel I should get but I don't see it yet. Perhaps I am neglecting properties of Brownian motion?

(in my MGF for Zt I assumed variance of Zt as 1. If, say, std deviation of then I get my result 1.

...UPDATED.

Got it! Checked again the definiton of Standard Brownian motion and the variance is t!!

So,

Now tell me if I was wrong!

PS Who would have thought studying maths could be such an emotional roller-coaster