# 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-$p$.

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I can't give you a bounty, because firt of all your answer is not suffitiently complete and also I've already obtained very complete answers in math.stackexchange.com/questions/302466/ and mathoverflow.net/questions/121744/… many days before you give yours. So it semms not so ethic of your part start a bounty just to obtain an arbitrage of points. It's the kind of thing I see just in a forum of finance, never in science one. –  Paul Feb 23 '13 at 18:39
@Paul 1) There doesn't exist any "arbitrage of points" 2) Bounty was not initiated by Christian 3) "It seems not so ethic" to silently crosspost your question and 4) If you cannot give the bounty, the Community can (and will) –  Alexey Kalmykov Feb 23 '13 at 19:21
@Paul I (the moderator) started the bounty. I offered my own points to get your question more exposure in the hopes that it would get answered. Therefore, I'm the one who decides how the bounty gets awarded. I'm not sure why you are giving Christian grief when he was only trying to help; neither one of us knew that you had cross-posted your question. –  chrisaycock Feb 24 '13 at 23:48
Sorry everybody, spetially Christian Fries, I totaly misunderstood the procedure to start a bounty and all. I'm ashamed guys. –  Paul Feb 25 '13 at 13:45