How to understand the following brownian integral using Fubini’s method?

I am a little bit stucked with the following integral process, using Fubini’s method, this is an intermediate step of short rate Merton Model.

$ \int_{t}^{T} W(s)ds=\int_{0}^{\hat {T}}ds\int_{0}^{s}dW(u)\=\int_{0}^{\hat {T}}dW(u)\int_{u}^{\hat {T}}ds\=\int_{0}^{\hat {T}}(\hat{T}-u)dW(u) $

My more specific question is how did the change of integration variables proceed, as the process described by above integration is not very intuitive to me.

Many thanks!

$$ \begin{align*} \int_0^T W(t), dt &{}= \int_0^T!!\int_0^t dW(u),dt \ &{}= \int_0^T!!\int_u^T dt, dW(u) \&{}= \int_0^T (T - u),dW(u) \&{}= TW(T) - \int_0^T u, dW(u) \end{align*} $$ however i am not sure if it is what you are asking for

引用自：https://quant.stackexchange.com/questions/27928