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I am comparing the following optimization problems:

Prob Space =(Omega, F_T, (F_t)_{t=1, ..T}, P). Let X be an adapted process.

I denote E[|F_t] as the conditional expectation given F_t.

1.Z_t(omega)= max{ E[X_s | F_t](omega) : s=t, ..,T}

2. Y_t(omega)=max{E[X_tau | F_t](omega): tau is stopping time in {t, ...T} }

I know (2.) is a well-studied problem: The process Y is Snell envelope.

My question is, does (1.) make sense? Will (1.) be Z supermartingale? I know that Y_t >=Z_t.

Do you know of any references regarding (1)?

Thank you in advance.

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# Stochastic Optimization

Can you offer guidance or do you also need help?

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