Quiz: Conditional Expectation and the Tower Property

1. The modern measure-theoretic definition characterises $\mathbb{E}[X \mid \mathcal{G}]$ (for $X \in L^1$, sub-σ-algebra $\mathcal{G} \subseteq \mathcal{F}$) as:

2. Let $\Omega = \{a,b,c,d\}$ with $\mathbb{P}$ uniform, $\mathcal{G} = \sigma(\{a,b\},\{c,d\})$, and $X(a)=1,\ X(b)=3,\ X(c)=0,\ X(d)=4$. What is $\mathbb{E}[X \mid \mathcal{G}](a)$?

3. Same setup as Q2. Which correctly verifies $\int_{\{c,d\}} \mathbb{E}[X \mid \mathcal{G}] \, d\mathbb{P} = \int_{\{c,d\}} X \, d\mathbb{P}$?

4. The tower property: if $\mathcal{H} \subseteq \mathcal{G} \subseteq \mathcal{F}$, then $\mathbb{E}[\mathbb{E}[X \mid \mathcal{G}] \mid \mathcal{H}] = \mathbb{E}[X \mid \mathcal{H}]$ a.s. What is the key step in the proof?

5. In the $L^2$ geometric interpretation, $\mathbb{E}[X \mid \mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\Omega, \mathcal{G}, \mathbb{P})$. The orthogonality condition states:

6. If $X$ is independent of the σ-algebra $\mathcal{G}$, what is $\mathbb{E}[X \mid \mathcal{G}]$?

7. In the companion notebook (cell 1), what does the output confirm about $\mathbb{E}[X \mid \mathcal{G}]$ on atom $\{a,b\}$?

8. A risk quant prices a Bermudan swaption with LSMC, regressing on three state variables. The true $\mathbb{E}[\text{continuation} \mid \mathcal{F}_t]$ depends on a fourth omitted variable. Which limitation does this illustrate?