If $X$ is a supermartingale.
I need an upper bound as:
$\mathbb{E}[\sup_{-\tau\leq\theta\leq0}\|X(\theta)\|^k]\le K \sup_{-\tau\leq\theta\leq0}\mathbb{E}[\|X(\theta)\|^k], $ where $\tau>0$ and $K$ is some constant.
Is there any way to prove such a result under some additional assumptions?