= Solution
Suppose the forward field is differentiable in maturity with the stochastic integrability needed for differentiation. As time increases, both arguments of $r_t=f(t,t)$ increase. The <diagonal short-rate dynamics in the Heath-Jarrow-Morton model> are therefore
$$
dr_t=[\partial_Tf(t,t)+\alpha^Q(t,t)]dt+\sigma(t,t)dW_t^Q.
$$
Under the HJM drift restriction from part (a), $\alpha^Q(t,t)=0$, giving
$$
\boxed{dr_t=\partial_Tf(t,t)\,dt+\sigma(t,t)dW_t^Q.}
$$
More explicitly the slope is
$$
\partial_Tf(t,t)=f_0'(t)+\int_0^t\partial_T\alpha^Q(s,t)\,ds
+\int_0^t\partial_T\sigma(s,t)\,dW_s^Q.
$$
It depends on the forward curve and potentially the whole history, not necessarily on $r_t$ alone. Without the stated maturity regularity, the original forward equations do not by themselves guarantee this classical <Itô process> formula for the diagonal.
\b[The <short rate> need not be <Markov>, even with one Brownian factor and deterministic volatility.] An explicit example of <One Brownian factor does not imply a Markov short rate> takes $f_0(T)=0$ and $\sigma(t,T)=T-t$. Its no-arbitrage forward drift is $\alpha^Q(t,T)=(T-t)^3/2$, so
$$
r_t=\frac{t^4}{8}+\int_0^t(t-s)dW_s
=\frac{t^4}{8}+I_t,\qquad I_t=\int_0^tW_sds.
$$
The continuous rate history determines $W_t$ from the left <derivative> of $I_t$. Hence for $h>0$,
$$
\mathbb E[I_{t+h}\mid\mathcal F_t^r]=I_t+hW_t.
$$
But $W_t$ is not determined by $I_t$: joint normality, $\operatorname{Var}I_t=t^3/3$, and $\operatorname{Cov}(W_t,I_t)=t^2/2$ give
$$
\operatorname{Var}(W_t\mid I_t)=t-\frac{(t^2/2)^2}{t^3/3}=\frac t4>0.
$$
Thus the conditional mean of the future given the whole rate history cannot be a function of the current rate alone, contradicting the <Markov property>. Enlarging the state to $(I_t,W_t)$ gives a two-dimensional <Markov process>; special HJM specifications can instead close on a single <short rate>.
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