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 ..?!?)