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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?

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