Assuming the usual setup of:
- $\left(\Omega, \mathcal{S}, \mathbb{P}\right)$ our probability space endowed with a filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\in[0,T]}$,
- $T>0$ denoting the option maturity,
- an $\mathbb{F}$-adapted process $Z=\left(Z_t\right)_{t\in[0,T]}$ modeling the discounted value of the option payoff at time $t$;
Why do we define the problem of pricing an American option as: $$ {\text{ess}\sup}_{\tau\in\mathrm{T}_{[0, T]}} \mathbb{E}\left[Z_{\tau}|\mathcal{F}_0\right] $$ and not as: $$ {\text{ess}\sup}_{s\in[0, T]} \mathbb{E}\left[Z_{s}|\mathcal{F}_0\right]? $$ In the above $\mathrm{T}_A$ is the set of all stopping times (with respect to our filtration $\mathbb{F}$) with values in the set $A$.
Non-mathematical common sense suggests that the option holder is basically only interested in a moment $s$ when to exercise the option optimally, so why should he be interested in optimizing over all stopping times?
My further doubts stem from the fact that every $s\in[0,T]$ is obviously also a stopping time, therefore we have an inclusion of the second formulation in the first and it would appear reasonable to state that:
$$ {\text{ess}\sup}_{s\in[0, T]} \mathbb{E}\left[Z_{s}|\mathcal{F}_0\right] \leq {\text{ess}\sup}_{\tau\in\mathrm{T}_{[0, T]}} \mathbb{E}\left[Z_{\tau}|\mathcal{F}_0\right]. $$
On the other hand, I am aware that finding the optimal stopping time provides the optimal moment of exercise, since our stopping times take values in $[0,T]$.
Therefore I have this uncomfortable feeling, that the first formulation might provide a bigger optimal value (because we are optimizing over a broader family of arguments) than the second, whereas I would imagine that both formulations should amount to the same result.
What sort of fallacies am I committing in following the presented line of thought?