Comparing Stochastic Optimization Problems: Z_t vs. Y_t

[submartingale]

Hi everyone,

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.

 

[Valkarie]

Yes, (1) does make sense and it will be a supermartingale. This is because it is the maximum of a set of conditional expectations and these are all supermartingales (and hence also submartingales). The fact that Y_t >= Z_t is easy to show using Jensen's inequality. I'm not aware of any specific references for (1), but you can find more general information on supermartingales in any textbook on stochastic processes.