$$ \int_A E[X|G](\omega)dP(\omega) = \int_A X(\omega)dP(\omega) $$ for all sets A belong to Sigma algebra G.
Quote from Steven Shreve’s Bk 2
- Say G is sigma algebra generated by some random variable W i.e. (G= σ(W))
- E[X|G] is G-measurable
This property ensures that E[X|G] is indeed an estimate of X. It gives the same averages as X over all the sets in G. If G has many sets, which provide a fine resolution of the uncertainty inherent in w (omega), then this partial-averaging property over the small sets in G says that the E[X|G] is good estimator of X. If G has only a few sets, this partial-averaging property guarantees only that E[X|G] is a crude of X. The above definition raises two immediate questions.
- First, does there always exist a random variable E[X|G] satisfying the above properties above.
- Second, if there is a random variable satisfying these properties is it unique?
The answer to the first question is yes, and the proof of the existence of E[X|G] is based on the Radon-Nikodym Theorem The answer to the second is a qualified yes, as explained below. Suppose Y and Z both satisfy the above two conditions. Because both Y and Z are G-measurable, their difference (Y – Z) as well, and thus the set A = {Y-Z > 0} is in G. From the above equation we have
$$ \int_A Y(\omega)dP(\omega) = \int_A X(\omega)dP(\omega) = \int_A Z(\omega)dP(\omega) $$ And thus $$ \int_A (Y(\omega) - Z(\omega))dP(\omega) = 0 $$
I am not clear what the above property wants to establish. Is there a way anyone can convey the above message in a simple language? Use a simple language to draw an analogy? Hope some of the experts help me get over this understanding gap! I cannot skip this topic as this repeats in Chapter 5 Girsanov’s theorem (Risk Neutral measure) Kindly guide/help me.
Also, many thanks to those who came up with the idea of this website/forum. It is so much easy to post a question with mathematical formula.
Thank you
In the textbook $\textbf{Probability Theory and Examples }$ 5th edition by Rick Durrett, on page 178, there is the definition of the conditional expectation and there is proof on existence and uniqueness property. You should take a deep look at this chapter before learning Martingale.