Integral question

[TNGL]

Can somebody please explain to me how the bit in yellow was obtained? I can't figure it out.

For clarity, Wu is a Wiener process.

 

[TNGL]

Highlighter probably.

I think we're getting closer to the answer...

 

[TNGL]

It's the integral of the final term, ie. after the second plus sign. Recall that the integral of e^-at is (1/a)(1-e^-at)

I have that bit but if you take the integral of the existing integral it's not as simple as that. In this cause we have e^-a(T-u) and so if you integrate that between T and 0 you won't end up with it an equation in the form you mention above.

I also cannot understand how it's gone from integrating between 0 and t with respect to Wu to integrating between 0 and T with respecting to Wu.

I'm not sure if I am failing to apply a really fundamental rule of calculus or if there is some result that is very specific to reaching the above integral that I am not aware of.

 

[TNGL]

In the first line extend the upper limit of the integral to infinity and replace the integrand with exp[-a(t-u)]H(t-u), where H is the Heaviside function. Then integrate your expression and interchange the order of integration on the final term (valid application of Fubini's theorem since the integrand is absolutely integrable with respect to the product measure dt dW(u)). This will yield the result after a couple of lines of computation.

Thanks for a serious answer!

I managed to solve this by using a different method in the end. Ultimately, I was making a mistake when changing the limits whilst changing the order of integration. By putting that step right, I was able to get the desired result.

 

[TNGL]

Potato.

But what does it mean

It's an interest rate model and by using the above result you can price bonds and other interest rate derivatives with it. However, it is too simplistic (compared to other models at least) to be used in practice .