Inequality of expectation of a submartingale in different time steps

Question

If we are given a submartingale $(X_n)_{n\geq 0}$ is it then always true that the expectation of $X_0$ is less than or equal to $X_n$, i.e. $$E[X_0] \leq E[X_n] ?$$ If this is so, I don't see how this follows from the definition of a submartingale, i.e. $$E[X_{n+1}|F_{n}]\geq X_n$$

Answer 1

A submartingale satisfies $$ \mathbb{E}[X_{n+1}\mid \mathcal{F}_n]\geq X_n $$ and therefore $$ \mathbb{E}[X_{n+2}\mid \mathcal{F}_n]=\mathbb{E}[\mathbb{E}[X_{n+2}\mid \mathcal{F}_{n+1}]\mid \mathcal{F}_n]\geq \mathbb{E}[X_{n+1}\mid \mathcal{F}_n]\geq X_n$$ and similarly by induction one can show that if $m\geq n$ then $$ \mathbb{E}[X_m\mid \mathcal{F}_n]\geq X_n$$ Then taking expectations yields $$ \mathbb{E}[X_m]\geq \mathbb{E}[X_n]$$ if $m\geq n$. In particular, $$ \mathbb{E}[X_n]\geq \mathbb{E}[X_0]$$ for all $n\geq 0$.