Double Stochastic Integral

Someone please explain how to simplify this integral directly if possible(!)
Shown on page 13 of the notes on Interest rate models.


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The double integral is simplified by swapping the order of integration.

In the form presented, we first integrate with respect to W(s), as s varies between t and u. Then we integrate with respect to u, as u varies between t and T.

So overall, this is covered by the relationships: t <= s <= u <= T

Swapping the order, we can take u first and very it between s and T. By doing this, we have a standard non-stochastic integral that we can work out directly. This leaves the integral with respect to W(s), where s now needs to vary between t and T.

This is a more familiar stochastic integral, whose mean and variance you should be happy working out.

 

Thanks!
It works, I had a problem with the limits of integration.
See the sketch:


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Great - glad it's sorted