= Solution
This part's conditional-expectation representation is its own hypothesis; it does not need the incorrect unrestricted claim in part (b). Fix $T$, and write
$$
I_t(T)=\int_t^Tf_t(u)\,du,\qquad
\beta_t(T)=\int_t^TB_t(u)\,du,\qquad
M_t(T)=U_te^{-I_t(T)}.
$$
The assumed representation makes $M(T)$ a <martingale>. The <stochastic Fubini theorem> gives
$$
dI_t(T)=\left(\int_t^TA_t(u)\,du-f_t(t)\right)dt+\beta_t(T)\,dW_t.
$$
Apply the <Itô formula> to $e^{-I}$ and use the equation for $U$ from part (a), including the cross-variation. The result is
$$
\frac{dM_t(T)}{M_t(T)}
=\left[f_t(t)-\frac18\sigma_t^2-\int_t^TA_t(u)\,du
+\frac12\beta_t(T)^2-\frac12\sigma_t\beta_t(T)\right]dt
+\left(\frac12\sigma_t-\beta_t(T)\right)dW_t.
$$
Uniqueness of the continuous <semimartingale> decomposition makes the drift vanish. Initially this is a $dt\,d\mathbb P$ statement; the assumed continuity in time and maturity extends it to the continuous versions simultaneously. Let $T\downarrow t$ to obtain
$$
\boxed{f_t(t)=\frac18\sigma_t^2=k\sigma_t^2.}
$$
Substitute back and differentiate the maturity integrals using their continuous integrands:
$$
\boxed{A_t(T)=B_t(T)\left(\int_t^TB_t(u)\,du-\frac12\sigma_t\right).}
$$
This is the <forward drift restriction for square-root stock claims>. The $-\sigma_t/2$ term comes from the product cross-variation and must be retained. For the uninformative zero-<stock> case, the representation does not identify $f$; as usual a positive initial <stock> price is understood.
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