TITLE: Show that the max of two submartingale is also a submartingale
QUESTION [7 upvotes]: Show that the maximum of two submartingales (relative to the same filtration) is a submartingale.

REPLY [16 votes]: By definition:
$$
\mathbb{E}[\max(X_n,Y_n)|\mathcal{F}_{n-1}]\geq \mathbb{E}[X_n|\mathcal{F}_{n-1}]\geq X_{n-1}
$$
similarly:
$$
\mathbb{E}[\max(X_n,Y_n)|\mathcal{F}_{n-1}]\geq \mathbb{E}[Y_n|\mathcal{F}_{n-1}]\geq Y_{n-1}
$$
Hence:
$$
\mathbb{E}[\max(X_n,Y_n)|\mathcal{F}_{n-1}]\geq  \max(X_{n-1},Y_{n-1})
$$