The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I don't know the meaning of "tradable asset". For example: in Brigo's "interest rate models" p39:
In this proof, $\mathbb{E}^T$ means the expectation under the forward measure( the numeraire is $T$-bond $P(t,T)$). $\mathbb{E}$ means the expectation under the risk-neutral measure (numeraire $B(t)=e^{\int_0^tr_s\,ds}$) I think he uses the risk-neutral pricing formula in the first equation: $$\mathbb{E}^T[r_T|\mathcal{F}_t]=r_t/P(t,T)$$ $$\mathbb{E}[e^{-\int_0^Tr_s\,ds}r_T|\mathcal{F}_t]=r_te^{-\int_0^tr_s\,ds}$$
my question is: 1. why the short rate $r_t$ is a tradable asset?
Let $H$ be the payoff function. In practice, I often apply the pricing formula as long as $H\in L^2$ regardless of whether it is a tradable asset or not. I know there is a complete market hypothesis: every derivative in the market is replicable.
my question is: 2 Does complete market implies $\forall H\in L^2$ is a tradable asset?
3 if the market is not complete, can we apply the pricing formula for $r_t$? i.e. how to judge $H$ is replicable or not?
