Comparing Stochastic Optimization Problems: Z_t vs. Y_t

In summary, the conversation discusses two optimization problems involving adapted processes. The first problem involves finding the maximum conditional expectation given a certain time, while the second problem involves finding the maximum conditional expectation for a stopping time within a given range. The second problem is well-studied and known as the Snell envelope. The first problem does make sense and will result in a supermartingale, with Y_t being greater than or equal to Z_t. While there are no specific references for the first problem, information on supermartingales can be found in textbooks on stochastic processes.
  • #1
submartingale
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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.
 
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  • #2
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.
 

FAQ: Comparing Stochastic Optimization Problems: Z_t vs. Y_t

1. What is stochastic optimization?

Stochastic optimization is a mathematical approach to solving complex problems that involve randomness or uncertainty. It uses probability and statistics to find the best solution among a large number of possible options.

2. What are some real-world applications of stochastic optimization?

Stochastic optimization has various applications in fields such as finance, engineering, and computer science. Examples include portfolio optimization, resource allocation, and machine learning algorithms.

3. How does stochastic optimization differ from traditional optimization methods?

Unlike traditional optimization methods, stochastic optimization takes into account the random and uncertain factors that may affect the outcome. It also allows for the optimization of non-differentiable or noisy objective functions.

4. What are some common techniques used in stochastic optimization?

Some common techniques used in stochastic optimization include gradient descent, simulated annealing, genetic algorithms, and Markov chain Monte Carlo methods.

5. What are the benefits of using stochastic optimization?

Stochastic optimization can lead to more robust and flexible solutions that can adapt to changing environments. It also allows for the incorporation of expert knowledge and can handle complex and non-linear problems more effectively.

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