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Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a supermartingale.

Can I say from the optional stopping theorem that I have :

$$E[Y_T 1_{A}| F_t] \leq E[Y_S 1_{A}| F_t]$$

I am guessing that this is true for $A \in F_S$, but I could not show it.

Any ideas ?

Thanks

Cal

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