Heath-Jarrow-Morton model ($df(t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)dudt+\sigma (t,T)d{W}_t$)

A model evolving the entire instantaneous forward rate curve. In the displayed one-factor risk-neutral measure dynamics the drift restriction makes every suitably integrable discounted zero-coupon bond price a martingale. The multi-factor form replaces the product of volatilities by their dot product.

In the one-factor Heath-Jarrow-Morton model, put $b_t^T={\int }_t^T\sigma (t,u)du$. The Itô formula gives $dP/P=r_{t}dt-b_t^{T}d{W}_t$. Thus multiplying by the discount factor yields the displayed stochastic exponential. Bounded forward volatility on a finite maturity horizon implies the Novikov condition, so the discounted bond is a true martingale.