Discounted bond price martingale ($D_{t}P(t,T)=P(0,T)\mathcal{E}(-{\int }_0^{\cdot }{b}_s^{T}d{W}_s{)}_t$)

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.