Derivation — The HJM Drift Condition from the Risk-Neutral Bond Price

The HJM drift condition states that $\alpha^Q(t,T) = \sigma(t,T)\cdot\int_t^T\sigma(t,u)\,du$ under the risk-neutral measure. This is not an assumption but a theorem — it follows entirely from the requirement that the discounted bond price be a local martingale under $Q$. The proof has three moving parts: (1) Itô's formula applied to the log bond price $\ln P(t,T) = -\int_t^T f(t,u)\,du$, (2) the stochastic Fubini theorem to interchange the maturity integral and the stochastic integral, and (3) the characterisation of the ELMM via the vanishing of the drift in the discounted bond price process. This derivation note assembles these three ingredients into a complete proof, then extends to derive the short-rate dynamics as a by-product.