I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to $$dX(t) = \alpha(t) dW^N(t)$$ wrt the numeraire $$N(t) = \frac{1}{P(0,t)} e^{H(t)X(t)+H^2(t)\int_0^t\alpha^2(s)ds}.$$ The functions $H$ and $\alpha$ are deterministic and can be chosen (almost) arbitrarily.

I would like to understand why $N$ is even eligible as a numeraire in the first place. It is positive, but without any other assumption I don’t see how it must be a tradeable asset.

Also the SDE for $X$ depends on $W^N$, which depends on $N$, which in turn depends on $X$. I understand that assuming everything is well defined and $N$ is a valid numeraire we can derive the explicit form under $Q$ by Girsanov, but isn’t the definition a bit circular?