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The value of an account that continuously reinvests at the short rate $r$, with $B_0=1$. Its reciprocal is the discount factor $D_t=B_t^{-1}$.

The continuously compounded rate inferred for an infinitesimal investment interval at future maturity $T$, as seen at time $t$. For a unit-face-value zero-coupon bond, $P(t,T)=\exp (-{\int }_t^{T}f(t,u)du)$ and the short rate is $f(t,t)$.

Write the discount factor as $D_t=B_t^{-1}=\exp (-{\int }_0^t{r}_{s}ds)$. Splitting the time integral at $t$ givesThe random variable $D_T$ lies in $(0,1]$ because the short rate is nonnegative and continuous on the finite maturity interval. A process of conditional expectations of an integrable terminal variable is a martingale, by the tower property of conditional expectation. Therefore

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.

For fixed maturity $T$, define $b_t^T={\int }_t^T\sigma (t,u)du$ and $I_t^T={\int }_t^{T}f(t,u)du$. The stochastic Fubini theorem and the moving lower endpoint giveThe factor $1/2$ is the integral over one of the two triangles in the square $[t,T{]}^2$. Applying the Itô formula to $P(t,T)=e^{-I_t^T}$, its quadratic-variation correction cancels that factor:Set $D_t=e^{-{\int }_0^t{r}_{s}ds}$, the reciprocal of the continuous-time bank account. The Itô product rule then givesIf $|\sigma |\le M$, then $|b_t^T|\le MT$ on this finite horizon. The Novikov condition holds, so this stochastic exponential is a true martingale, not merely a local martingale. The discounted price is thereforeThe authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future short rates and is incorrect here.