Smoothing

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable on $\Omega$. Suppose $\mathcal{G}{1} \subset \mathcal{G}{2} \subset \mathcal{B}$ are sub-Sigma Algebras on $\Omega$. Then

$$\mathbb{E}\big[ \mathbb{E}[X|\mathcal{G}{2}] | \mathcal{G}{1} \big] = \mathbb{E}[X| \mathcal{G}_{1}]$$

Recalling that Conditional Expectation over the Trivial Sigma Algebra is Expectation, we see that if $\mathcal{G}_{1}$ is the Trivial Sigma Algebra on $\Omega$ then

$$\mathbb{E}\big[ \mathbb{E}[X|\mathcal{G}{2}] | \mathcal{G}{1} \big] = \mathbb{E}[X| \mathcal{G}_{1}] = \mathbb{E}[X]$$

We show that $\mathbb{E}[X| \mathcal{G}{1}]$ satisfies the properties for Conditional Expectation of $\mathbb{E}[X|\mathcal{G}{2}]$ with respect to $\mathcal{G}{1}$. By definition of Conditional Expectation, we know $\mathbb{E}[X|\mathcal{G}{1}]$ is $\mathcal{G}{1}$-measureable. Let $A \in \mathcal{G}{1}$. Then

$$\int\limits_{A} \mathbb{E}[X|\mathcal{G}{1}] = \int\limits{A} X = \int\limits_{A} \mathbb{E}[X|\mathcal{G}_{2}]$$

where the first equality follows from the definition of Conditional Expectation and the second because $A \in \mathcal{G}_{2}$ as well. Thus,

$$\mathbb{E}\big[ \mathbb{E}[X|\mathcal{G}{2}] | \mathcal{G}{1} \big] = \mathbb{E}[X| \mathcal{G}_{1}]$$

$\blacksquare$

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable on $\Omega$. Suppose $\mathcal{G}{1} \subset \mathcal{G}{2} \subset \mathcal{B}$ are sub-Sigma Algebras on $\Omega$. Then

$$\mathbb{E}\big[ \mathbb{E}[X|\mathcal{G}{1}] | \mathcal{G}{2} \big] = \mathbb{E}[X| \mathcal{G}_{1}]$$

Since $\mathcal{G}{1} \subset \mathcal{G}{2}$ and $\mathbb{E}[X|\mathcal{G}{1}] \in \mathcal{G}{1}$, we have that $\mathbb{E}\big[ \mathbb{E}[X|\mathcal{G}{1}] | \mathcal{G}{2} \big] = \mathbb{E}[X| \mathcal{G}_{1}]$ because Conditioning on known information is Idempotent. $\blacksquare$