I was reading up on submartingales and supermartingales and saw this statement which I do not understand.
A stochastic process $(X_n)_{n\geq 1}$ is a submartingale with respect to a filtration $(\mathcal{F}_n)_{n\geq 1}$ if and only if $(-X_n)_{n\geq 1}$ is a supermartingale with respect to $(\mathcal{F}_n)_{n\geq 1}$.
Need some explanation on this statement. Don't really understand the meaning behind the statement
A process $(X_n)_{n\geq 1}$ is a sub/super-martingale with respect to $(\mathcal{F}_n)$ if
$(X_n)_{n\geq 1}$ is adapted to $(\mathcal{F}_n)_{n\geq 1}$
$X_n$ is integrable for every $n\geq 1$
${\rm E}[X_{n+1}\mid\mathcal{F}_n]\geq X_n$ (sub) or ${\rm E}[X_{n+1}\mid\mathcal{F}_n]\leq X_n$ (super) for every $n\geq 1$.
Assume that $(X_n)_{n\geq 1}$ is a sub-martingale. Then we have to argue that $(-X_n)_{n\geq 1}$ is a super-martingale. It clearly satisfies 1 and 2 (why?). As for the last item, we just have to show that for all $n\geq 1$: $$ {\rm E}[-X_{n+1}\mid\mathcal{F}_n]\leq -X_n $$ using that we know ${\rm E}[X_{n+1}\mid\mathcal{F}_n]\geq X_n$. This is pretty obvious, isn't it?