= Solution
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 give
$$
dI_t^T=
\left[-r_t+\int_t^T\sigma(t,u)\int_t^u\sigma(t,v)\,dv\,du\right]dt
+b_t^T\,dW_t
=\left[-r_t+\frac12(b_t^T)^2\right]dt+b_t^T\,dW_t.
$$
The 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:
$$
\frac{dP(t,T)}{P(t,T)}=r_t\,dt-b_t^T\,dW_t.
$$
Set $D_t=e^{-\int_0^t r_sds}$, the reciprocal of the <continuous-time bank account>. The <Itô product rule> then gives
$$
D_tP(t,T)=P(0,T)\exp\!\left[-\int_0^t b_s^T\,dW_s-\frac12\int_0^t(b_s^T)^2ds\right].
$$
If $|\sigma|\leq M$, then $|b_t^T|\leq MT$ on this finite horizon. The <Novikov condition> holds, so this <stochastic exponential> is a true <martingale>, not merely a <local martingale>. \b[The discounted price is therefore]
$$
\boxed{D_tP(t,T)\text{ is a }\mathbb Q\text{-martingale}.}
$$
The authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future <short rates> and is incorrect here.
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