= Solution
Under the specified <risk-neutral measure>, the <Itô product rule> and $d(B_t^{-1})=-r_tB_t^{-1}dt$ give
$$
d(B_t^{-1}P(t,T))=B_t^{-1}P(t,T)\Sigma(t,T)dW_t.
$$
Thus discounted traded <zero-coupon bonds> have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true <martingales>, not merely <local martingales>. If $X$ is an attainable <financial payoff> with $\mathbb E_Q|B_T^{-1}X|<\infty$, its <self-financing portfolio> value $V_t$ has terminal value $V_T=X$ and satisfies
$$
B_t^{-1}V_t=\mathbb E_Q[B_T^{-1}X\mid\mathcal F_t].
$$
Multiplying by $B_t$ yields
$$
\boxed{V_t=\mathbb E_Q\left[\exp\left(-\int_t^Tr_sds\right)X\mid\mathcal F_t\right].}
$$
This proves the requested conditional pricing formula. In a <Brownian filtration>, the <Martingale representation theorem> realizes the discounted conditional <expectation> as a <stochastic integral>. If an available bond has nonzero <Itô diffusion> exposure, its holdings can match that integral, with the remaining value in the <bank account>, giving replication under the usual square-integrability conditions.
These attainability or <market completeness> hypotheses matter for a completely arbitrary <contingent claim>: existence of a <risk-neutral measure> alone need not imply a unique price for every unspanned <financial payoff>, and a zero-drift <local martingale> alone need not satisfy the displayed terminal <expectation> identity. In the general path-dependent <Heath-Jarrow-Morton model>, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current <short rate> is a sufficient state for the <financial payoff> and dynamics.
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