Why is the martingale factor a martingale in Hansen's 2012 Dynamic Valuation Decomposition?
The idea is that after the Perron-Frobenius problem is solved, you have $e$ such that
\begin{align}
E \left[\frac{M_{t+1}}{M_t} e(X_{t+1}) \middle | X_t = x \right]
 &amp;= e(x) \exp(\eta) \\
E \left[\frac{M_{t+1}}{M_t} \frac{e(X_{t+1})}{e(X_t)} \exp(-\eta) \middle | X_t = x \right]
 &amp;= 1 \\
E \left[\frac{\widetilde M_{t+1}}{\widetilde M_t} \middle | \mathcal F_t \right]
 &amp;= 1 \\
E \left[\widetilde M_{t+1} \middle | \mathcal F_t \right]
 &amp;= \widetilde M_t.
\end{align}

Thus we see that $\widetilde M_t$ is a martingale.