# Upperbound on expectation of supremum

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?