I have a process$$X(t,\omega)=\cases{0,\quad \text{if } t<\tau(\omega) \\ 1\quad \text{if } t\ge\tau(\omega)}$$
where $\tau$ is a stopping time. I need to show that $X(t, \omega)$ is a submartingale. Can anyone help, especally with the 3 condition i.e for $s\le t$ we have
$$X(s)\le E(X(t)| F(s)) $$ where $F(s)$ is a filtration generated by process $X$.
Note that $X(t)\geq X(s)$. Once you show that $X(t)$ is adapted (which is a good exercise in stopping times), this gives you that $X(s)$ is $F(s)$-measurable and hence,
$$ E(X(t)|F(s))\geq E(X(s)|F(s))=X(s) $$ almost surely.