Are these true for a martingale? $E\left[ \frac{X_{n+1}}{X_n} \right] = 1, E\left[ \frac{X_{n+2}}{X_n} \right] = 1 $

Question

Let $(X_n)_{n \in \mathbb{Z}_+}$ be a martingale, $X_n(\omega) \neq 0$, and $X_{n+1}/X_n, X_{n+2}/X_n \in L^1 (n \in \mathbb{Z}_+)$

Do the following hold for $n \in \mathbb{Z}_+$?

$E\left[ \frac{X_{n+1}}{X_n} \right] = 1, E\left[ \frac{X_{n+2}}{X_n} \right] = 1 $

Answer 1

$E(X_{n+m}|\mathcal F_m)=X_m$. This implies $E(\frac {X_{n+m}} {X_m}|\mathcal F_m)=1$ provided $\frac {X_{n+m}} {X_m}$ is integrable. Taking expectation on both sides we get $E(\frac {X_{n+m}} {X_m})=1$ .