Bounding a Snell Envelope with a Conditional Maximum

Summary

The document asks how to bound the expected supremum of powers of a Snell envelope, defined through conditional expectations of a nonnegative process, by the corresponding quantity for the conditional expectation of the process maximum. It provides a brief proposed justification: the Snell envelope is bounded by that conditional maximum, so taking suprema and then powers should preserve the inequality when the power is at least one.

The response is tentative and does not establish the key point that the envelope is bounded by the process bar X_t = E[bar X | F_t]. Nor does it spell out conditions needed for the supremum, integrability, or the claimed power inequality. The excerpt therefore highlights a useful stochastic-process comparison question, but its answer is not a complete proof and should be checked against the precise assumptions and definitions before relying on it. It concerns mathematical tools rather than a trading strategy or empirical market result.

Key ideas

• The question compares the maximal Snell envelope with a conditional expectation of the process maximum.
• The proposed argument relies on bounding the envelope pointwise by the conditional maximum process.
• Taking suprema and powers would preserve the bound under the stated power restriction if the pointwise comparison holds.
• The response is explicitly uncertain and does not provide a rigorous derivation or verify all required assumptions.

Tags

# Upper bound concerning Snell envelope


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

## Answer by Christian Fries (score 3)

https://quant.stackexchange.com/a/7327

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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