Consider the stochastic differential equation,
, for all ,
where is a standard Brownian motion, and are positive constants. Derive the unique solution to the above SDE . Calculate and .
Skipping working the unique solution is...
.
Now to find the expectation and variance. I've realized that a lot of the things I do I do because I have seen them in lectures or done them in other questions but I now realize I have no idea why they are true.
. Now why is that last step true?
My guess is something to do with...
But I don't know what to do with the although I assume that it eventually comes out to be .
Now the variance...
The way my lecturer does it is to calculate but I always calculate instead then do the usual subtraction of .
So, I get...
.
but now I am not sure how to justify the next step.
(actually should that be ..?!?)