Diagonal short-rate dynamics in the Heath-Jarrow-Morton model ($d{r}_t=[{\partial }_{T}f(t,t)+\alpha (t,t)]dt+\sigma (t,t)d{W}_t$)

For a maturity-differentiable Heath-Jarrow-Morton model forward field with the integrability needed for stochastic differentiation, the short rate $r_t=f(t,t)$ has the displayed dynamics. Indeed changing observation time uses the forward-rate stochastic differential equation, while changing maturity adds ${\partial }_{T}f(t,t)dt$. Under a risk-neutral measure, $\alpha (t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so $\alpha (t,t)=0$. The remaining drift depends on the diagonal slope of the whole forward curve, which need not be determined by $r_t$; a Markov process for the short rate is an additional modelling restriction. Without sufficient maturity regularity, the diagonal need not admit this classical Itô process formula.