# Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $\mathbb E \left \{ \bar X \right\}< \infty$ (where $\bar X =\sup _{0\leq t \leq T} X_t$) and its Snell envelope

$$\hat {X_\theta} = \underset {\tau \in \mathcal T _{\theta,T} } {\text{ess sup}} \ \mathbb E \left\{ X_\tau | \mathcal F_\theta \right \}$$

I'd like to understand how justify the following inequality:

$$\mathbb E \left\{ \sup_{0\leq t \leq T} \hat X_t^p\right \} \leq \mathbb E \left\{ \sup_{0\leq t \leq T} \bar X_t^p\right \}$$

where $\bar X_t = \mathbb E \left\{ \bar X | \mathcal F_t \right \}$

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Can you please define the set below ess sup. Is the the set of all stopping times? Also, what is p? Do we have $p \geq 1$. –  Christian Fries Feb 19 '13 at 12:17
–  Alexey Kalmykov Feb 23 '13 at 18:13

I am not sure (had only a quick look), but isn't it that we have $\hat{X} \leq \bar{X}$ and hence we have the same for the $sup$ and given that $p \geq 1$ we have this for the power-of-$p$.