 Solution

Let $D_s=\exp (-{\int }_0^s{R}_{u}du)$, 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 isThe positive factor $D_s$ is measurable with respect to the current filtration. Dividing it out givesConversely, 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 $r\le s$, 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.