# Step by step integration of the Hull-White SDE

I'm struggling to understand the integration process of the Hull-White equation:

$$$$dr(t)=[\nu(t)-ar(t)]dt+\sigma dW(t)$$$$

In the majority of the references that I have consulted, apparently is trivial to integrate between the spot time s and the valuation time t reaching the following mean and variance results:

$$$$E[r(t)]=r(s)e^{-a(t-s)}+\alpha(t)-\alpha(s)e^{-a(t-s)}$$$$

$$$$Var[r(t)]=\dfrac{\sigma^2}{2a}[1-e^{-2a(t-s)}]$$$$

Where $$\alpha$$ is related to the forward rate $$f^{m}$$ and is defined as:

$$$$\alpha(t)=f^{m}(0,t)+\dfrac{\sigma^2}{2a}[1-e^{-at}]^{2}$$$$

Does anybody now a website or paper where all this integration process is explained step by step?

I have tried to look for it but in the best cases I have found just one or two intermediate steps without detailed explanation.

• Have a look here, particularly the section titled "Why not to use Short-hand notation" where a solution to the SDE is shown step by step. Commented Apr 9 at 18:22
• An excellent book covering a lot of useful stuff when it comes to modelling the Hull-White process, and a lot of other models, is the second volume of the trilogy "Interest Rate Modelling" by L. Andersen and V. Piterbarg. This might be a little advanced if you are very new to the topic though. Commented Apr 10 at 11:15

Start with $$d(e^{(at)}r(t))=e^{at}(-ar(t)dt+vol*dW(t)+ar(t)dt+v(t)dt)$$