Snell envelope and optimal stopping time Suppose $(G_n)_{0\leq n\leq N}$ is a process adapted to a filteing $(\mathcal{F}_n)_{0\leq n\leq N}$.
The Snell envelope of $(G_n)$ is the smallest supermatingale dominates $(G_n)$. It's defined as follows:$$S_N=G_N,S_{n}=\max\{G_n,E[S_{n+1}|\mathcal{F}_n]\},n=N-1,\ldots,0.$$
The stopping time $\tau=\inf_k\{0\leq k\leq N:S_k=G_k\}$ is the optimal stopping time to maximize the gain $EX_\tau$.
It can also be shown that $S_{n\wedge\tau}$ is a martingale, my question is, what is the significance of this result? We have already find the optimal rule( $\tau$), and we can compute the optimal gain $EX_\tau=ES_0$, so what can we say about the conclusion that $S_{n\wedge \tau}$ is a martingale?
 A: I'll assume that you're using $X$ and $G$ interchangeably.
The reason is that, to prove that for the stopping time $\tau^*$ is optimal (such that $E [G_{\tau^*}] = sup_{\tau \in [0, T]} E [G_\tau]$), both conditions need to be satisfied:


*

*$S_{\tau^*} = G_{\tau^*}$ a.s., and

*that the stopped sequence $S_{n \wedge \tau^*}$ is a martingale.


Since $S_n$ dominates $G_n$, the first condition only tells us that the process $G_n$ for $n \geq \tau^*$ is a supermartingale. 
This but tells us nothing about the process $G_n$ for $n \lt \tau^*$, except that we know that it is dominated by $S_n$. Since $E[S_{n}] \geq E[S_{\tau^*}]$ for $n \lt \tau^*$, it is entirely possible that $E[G_{n} \gt E[G_{\tau^*}]$ for some $n \lt \tau^* $, which would mean that ${\tau^*}$ is not optimal. 
But, if the stopped process $S_{n \wedge \tau^*}$ for $\forall n \in [0, T]$ is a martingale, then, for $n \lt \tau^*$, $E[S_{n}] = E[S_{\tau^*}]$ and, by the fact that $G_n$ is dominated by $S_n$, $E[G_{n}] \leq E[G_{\tau^*}]$. This confirms that $\tau^*$ is optimal.
Please note that this only proves that $\tau^*$ is an optimal stopping time, not that it is the smallest optimal stopping time.
A: This can be used when pricing an American option. The Snell envelope property guarantees the buyer of the option that this is the best possible price. But nobody would by a supermartingale. The second property gives the buyer the guarantee that the price is fair if he exercises optimally. 
