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I have troubles understanding the Tower Propery in my course on Introductory Mathematical Finance.

Basically I have: $$E(E(X|F_{t+1})|F_t)=E(X|F_t)$$ where $F_t \subseteq F_{t+1}$

$F_t$ is the information set at time $t$ and $F_{t+1}$ is the information set at time $t+1$.

I understand somehow the concept of measurability, but I do not understand why this equation holds. I would really appreciate if you could dumb it down as much as possible.

Thanks!

user91991993
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  • Are you familiar with the tower property for simpler cases, say, the case of discrete random variables? – angryavian Mar 17 '17 at 00:21
  • perhaps helpful: http://math.stackexchange.com/questions/41536/intuitive-explanation-of-the-tower-property-of-conditional-expectation – Badam Baplan Mar 17 '17 at 00:23
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    Intuitively, you're asking yourself at time $t$ "what do I expect I will expect $X$ to be at time $t+1$." If the answer were anything different from what you expect now at time $t$, you would be expecting something else now, now wouldn't you? – spaceisdarkgreen Mar 17 '17 at 00:35
  • Consider you are walking in a tree diagram, and you are starting at the root. At the root, you can calculate the unconditional expectation of your destiny, i.e. the expectation conditioned on the trivial sigma algebra which provide no further information to you. When you randomly selected a branch and moved forward by one step, you can calculate the conditional expectation, given you have selected this branch. You can calculate the conditional expectation for each branch similarly, so $E[X|\mathcal{F}_1]$ will be a random variable with support consists of these conditional expectation. – BGM Mar 17 '17 at 04:14
  • So if you take the expectation of this random variable, you will back to the unconditional expectation. And this results holds for each node. Another example is that you can think of taking the average of some quantity inside a population. It is equivalent to first you partitioned the whole population into several sub-population, and then take the average inside each sub-population, and finally take an overall average over all these averages. – BGM Mar 17 '17 at 04:18

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