I seek a basic form (SDE) to understand the Ho-Lee model.
I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example,
\begin{align*} dX_t &= -1/2 \alpha X_t dt + \sigma dWt, \\ r_t &=f(t,X_t)=(X_t)^2. \end{align*} Then, $f_t(t,x)=0$, $ f_x(t,x)=2x$ and $f_{xx}(t,x)=2$. By Itô's Lemma,
\begin{align*} dr_t&= \left(-1/2 \alpha X_t \cdot 2X_t+ 1/2\sigma^2 \cdot 2\right) dt+2\sigma X_t dW_t \\ &= \left(\sigma^2 -\alpha r_t\right)dt + 2\sigma \sqrt{r_t} dW_t. \end{align*}
So, what about the Ho-Lee model?
I know that the instantaneous forward is given by $$f(t, T) = f(0,T) + \sigma^2 (Tt - 1/2t^2) + \sigma W_t.$$
The short rate is optained using $r_t=f(t,t)$, that is:
$$r_t= r_{0} + 1/2 \sigma^2 t^2 + \sigma W_t.$$
But is there a way of defining the model via an SDE?