Step by step integration of the Hull-White SDE

Question

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

\begin{equation} dr(t)=[\nu(t)-ar(t)]dt+\sigma dW(t) \end{equation}

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:

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

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

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

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

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.

Answer 1

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

Integrate both sides with limits. That's all there is.

Forward is the expectation of the spot, once you have the spot from here.