= Solution
Let $D_s=\exp(-\int_0^sR_u\,du)$, where $R_u=F_{u,u}$ is the <short rate>. At maturity the <zero-coupon bond> pays one, so $Z_{T,T}=D_T$. If the discounted bond price is a <martingale>, its terminal <conditional expectation> is
$$
D_sP_{s,T}=Z_{s,T}=\mathbb E[D_T\mid\mathcal F_s].
$$
The positive factor $D_s$ is measurable with respect to the current <filtration>. Dividing it out gives
$$
\boxed{P_{s,T}=\mathbb E\left[\exp\left(-\int_s^TR_u\,du\right)\middle|\mathcal F_s\right].}
$$
Conversely, this identity says $Z_{s,T}=\mathbb E[D_T\mid\mathcal F_s]$. The <tower property> gives $\mathbb E[Z_{s,T}\mid\mathcal F_r]=Z_{r,T}$ for every $r\leq s$, establishing the <martingale> condition. The integrated <short rate> is a <Gaussian random variable> with finite <variance>, so the <Gaussian moment-generating function> proves that $D_T$ is <integrable>. Together with the preceding subparts, this proves all three descriptions equivalent.
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