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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.

The short rate is the limiting instantaneous forward rate at the present maturity. The continuously compounded zero-coupon bond price is obtained by integrating the instantaneous forward rate curve in its maturity variable. Both requested relations areIn particular $P(T,T)=1$, and $f(t,T)=-{\partial }_T\log P(t,T)$. The short rate here is instantaneous, rather than the one-period rate used in a discrete-time bank account.

Since $P(T,T)=1$, part (b) shows that the density process of the T-forward measure isIts terminal expectation is one and it is strictly positive, so the stated Radon-Nikodym derivative defines an equivalent probability measure. By the Girsanov theorem,is a Brownian motion under ${\mathbb{Q}}_T$. Substitution of $d{W}_t=d{W}_t^T-b_t^{T}dt$ cancels the entire instantaneous forward rate drift:Because the integrand is deterministic and bounded, its Itô integral is square-integrable on $[0,T]$. With the usual fixed initial instantaneous forward rate curve, the requested true martingale is

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.