Show that the max of two submartingale is also a submartingale

Question

Show that the maximum of two submartingales (relative to the same filtration) is a submartingale.

Answer 1

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:

$$ \mathbb{E}[\max(X_n,Y_n)|\mathcal{F}_{n-1}]\geq \max(X_{n-1},Y_{n-1}) $$