I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options.

I know that in general, the expectation of the supremum is $\geq$ the supremum of the expectation, but that there are special cases where the equality holds. My issue is that I saw a few proofs for the equality holding, but they either involve supremum over deterministic values or using 'policies' such as in stochastic control. So since the snell envelope, involves supremum over stopping times, which are random variables, I am unsure if they can apply here.

So I am trying to compare $\mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right)$ and $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, for some payoff process $Z_t$. I haven't been able to find any source that mentions this specific issue, and can't tell how to show whether they are equal. It seems to me like the optimal stopping time is just like a policy, so in that case the result from stochastic control can be used, but I am skeptical since I'd prefer a proof written out.

In a general sense, I know that the Snell Envelope is optimizing the Expectation of the payoff over all acceptable stopping times (random variables that don't look into the future). On the other hand, the expectation of the supremum is taking the expectation of the payoff after it has been optimized over these stopping times. Although I get the syntactical difference between the two, I can't seem to pinpoint an example of where they differ or how to even calculate the supremum of the payoff without using information from the future (to calculate the supremum of a random variable over multiple stopping times it seems like we are choosing the best stopping time based on the realization of the random variable).

I have seen a proof similar to the follows: (but I am not sure if it is valid or if it applies to stopping times)

  1. $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right) \leq \mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right)$ follows due to reasoning such as in: https://math.stackexchange.com/questions/2230255/supremum-of-expectation-le-expectation-of-supremum

  2. Let $\tau^*$ be the stopping time such that $\sup_{t\le\tau\le T} Z_\tau= Z_{\tau^*}$, then $\mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right) = \Bbb E\left(Z_{\tau^*}\mid \mathcal F_t\right) \leq \sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, where the last inequality follows since $\tau^*$ is a stopping time contained in the set of all admissible stopping times $t\le\tau\le T$

  3. So combining the two inequalities $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right) \leq \mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right)$ & $\mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right) \leq \sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, we have that $\mathbb E\left(\sup_{t\le\tau\le T}Z_\tau\mid \mathcal F_t\right) =\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$

I know that in general, when doing Monte-Carlo type algorithms for pricing these options, the Snell envelope approach is taken, where the Snell Envelope is turned into a dynamic programming problem, with no mention of the supremum being taken over the payoff itself and then calculating the expectation instead. Was hoping for some clarification on my confusion here. Thanks!

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    $\begingroup$ Under certain condition, there is a stopping time $\tau*$ such that $\sup_{t \le \tau \le T} \mathbb{E}(Z_{\tau} \mid \mathcal{F}_t) = \mathbb{E}(Z_{\tau*} \mid \mathcal{F}_t) $. Then you can show the equality. $\endgroup$ – Gordon Oct 3 '19 at 13:21
  • $\begingroup$ Can you elaborate on the condition? I want to know if this condition is valid during typical American option pricing scenarios. In typical stochastic control, it seems like the argument you just made is also applied to show the equality, but there doesn't seem to be mention of a condition. $\endgroup$ – Slade Oct 3 '19 at 13:36
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    $\begingroup$ May be later. But you have a look of Appendix D in the book Methods of Mathematical Finance. $\endgroup$ – Gordon Oct 3 '19 at 13:50
  • $\begingroup$ Okay, thanks! Just to clarify, I know that the supremum and expectation can be interchanged if we are dealing with $Z_{\tau}$ as the snell envelope of some other process $Y$, but in my case $Z$ is the payoff itself. I know there's a proof for the former case in that book. I'll edit my question in regards to your first comment. $\endgroup$ – Slade Oct 3 '19 at 15:19

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