your reasoning up to the last line is fine.
I have not seen the result you use on the last line so i can comment on that.
I need to find , where is a standard Brownian motion.
I tried the following.
Brownian motion has independent increments, and I should be able to write the above sum as two independent standard normal variables, i.e.
, where and are independet standard normal variables.
In the last line I used the fact that independent random variables have factoring moment-generating function and also the fact that .
Is the above reasoning correct, if not, what is wrong? Thanks!