= Solution
Since $d(B_t^{-1})=-r_tB_t^{-1}dt$, multiplying the given bond dynamics by $B_t^{-1}$ cancels the drift and gives $dZ_t=Z_t\Sigma(t,T)dW_t^Q$. For an integrable discounted terminal payoff, the risk-neutral value process is the discounted-payoff <martingale>
$$
M_t=\mathbb E_Q[B_T^{-1}X\mid\mathcal F_t],\qquad V_t=B_tM_t.
$$
For an attainable claim this also follows directly from its self-financing discounted wealth being a true <martingale>, with terminal value $B_T^{-1}X$. In a Brownian market, the <Brownian martingale representation theorem> provides $dM_t=\psi_tdW_t^Q$; a traded bond with nonzero diffusion exposure can hedge it by holding $\psi_t/(Z_t\Sigma_t)$ bond units, with the remainder in the <bank account>. Thus, under the usual attainability and integrability assumptions, this construction gives the replicated price as well as the <risk-neutral pricing> formula.
Because $B_t$ is $\mathcal F_t$-measurable,
$$
\boxed{V_t=\mathbb E_Q\left[\exp\left(-\int_t^Tr_sds\right)X\,\middle|\,\mathcal F_t\right].}
$$
In a general <Heath-Jarrow-Morton model>, writing this value as $V(t,r_t)$ requires an additional state-reduction assumption. The <conditional expectation> is valid without asserting that the <short rate> alone is the full state.
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