9 — Conditional Expectation
We work on a probability space
\((\Omega, \mathcal{F}_0, P), \qquad \mathcal{F} \subseteq \mathcal{F}_0.\)
For any integrable random variable $X$ and event $A \in \mathcal{F}$: \(E(X \cdot \mathbf{1}_A) = E\big(E_{\mathcal{F}}(X)\cdot \mathbf{1}_A\big).\)
We require:
- $E_{\mathcal{F}}(X) \in \mathcal{F}$
- $E(X\mathbf{1}A) = E(E{\mathcal{F}}(X)\mathbf{1}_A)$ for all $A \in \mathcal{F}$
This defines $E_{\mathcal{F}}(X)$, the conditional expectation of $X$ given $\mathcal{F}$.
Example: Independent Variables
Assume $X, Y$ are independent.
Let $\varphi(X,Y):\mathbb{R}^2 \to \mathbb{R}$ be measurable.
Let $\mathcal{F} = \sigma(X)$.
Claim:
\(E(\varphi(X,Y)\mid \mathcal{F}) = g(X),\)
where
\(g(x) = E[\varphi(x,Y)] = \int_{\mathbb{R}} \varphi(x,y)\, dF_Y(y).\)
Verification: \(E(\varphi(X,Y)\mathbf{1}_A) = \int_{x\in A} \left( \int_{\mathbb{R}} \varphi(x,y)\, dF_Y(y) \right) dF_X(x) = E(g(X)\mathbf{1}_A).\) So $E_{\mathcal{F}}(\varphi(X,Y)) = g(X)$.
Properties of Conditional Expectation
1. Linearity
\(E_{\mathcal{F}}(aX+bY) = aE_{\mathcal{F}}(X) + bE_{\mathcal{F}}(Y).\)
2. Monotonicity
If $Y \ge X$, then \(E_{\mathcal{F}}(Y) \ge E_{\mathcal{F}}(X).\)
3. Monotone Convergence (MCT)
If $X_n \uparrow X$ and $X_n \ge 0$, and $E(X)<\infty$, then \(E_{\mathcal{F}}(X_n) \xrightarrow{a.s.} E_{\mathcal{F}}(X).\)
4. Jensen’s Inequality
If $\varphi$ is convex and $E|\varphi(X)|<\infty$, then \(\varphi(E_{\mathcal{F}}(X)) \le E_{\mathcal{F}}(\varphi(X)) \quad a.s.\)
Application: contraction in $L^p$
If $X\in L^p$ for $p\ge1$, then \(\|E_{\mathcal{F}}(X)\|_p \le \|X\|_p.\)
5. Tower Property
If $\mathcal{F}_1 \subseteq \mathcal{F}_2$, then: \(E_{\mathcal{F}_1}(E_{\mathcal{F}_2}(X)) = E_{\mathcal{F}_1}(X), \qquad E_{\mathcal{F}_2}(E_{\mathcal{F}_1}(X)) = E_{\mathcal{F}_1}(X).\)
6. Pulling Out Known Factors
If $X$ is $\mathcal{F}$-measurable and $E|XY|<\infty$, then: \(E_{\mathcal{F}}(XY) = X\,E_{\mathcal{F}}(Y).\)
Proof idea: for all bounded $Z\in\mathcal{F}$, \(E(E_{\mathcal{F}}(XY)Z) = E(XYZ) = E(XE_{\mathcal{F}}(Y)Z).\)
7. Conditional Expectation is the $L^2$ Projection
(Pythagorean theorem in $L^2$)
Assume $E|X|^2 < \infty$.
Then $E_{\mathcal{F}}(X)$ is the orthogonal projection of $X$ onto the subspace $L^2(\mathcal{F})$.
For any $Y\in L^2(\mathcal{F})$: \(\|X - E_{\mathcal{F}}(X)\|_2^2 \le \|X-Y\|_2^2.\)
Geometric interpretation (shown in page 2 diagram):
- $X$ is a point in Hilbert space
- $L^2(\mathcal{F})$ is a subspace
- $E_{\mathcal{F}}(X)$ is the closest point to $X$ in this subspace
Additional Notes from the Lecture
Doob’s Interpretation
For a measurable function $g$, \(E_{\mathcal{F}}(g(X))(\omega) = \int_{\mathbb{R}} g(\alpha)\, d\mu_\omega(\alpha),\) where $\mu_\omega$ is a probability measure depending on $\omega$.
Conditional Probability
For $A \in \mathcal{F}_0$: \(P_{\mathcal{F}}(A)(\omega) = E_{\mathcal{F}}(\mathbf{1}_A)(\omega).\)
Properties:
- $P_{\mathcal{F}}(A \cup B) = P_{\mathcal{F}}(A) + P_{\mathcal{F}}(B)$ (disjoint $A,B$)
- If $A_i$ disjoint, \(P_{\mathcal{F}}\Big(\bigcup_i A_i\Big) = \sum_i P_{\mathcal{F}}(A_i).\)
Doob’s Focus
The handwritten note at the bottom (page 3) says:
“What was the trick of Doob?
Focus on the real line.”
This refers to Doob representing conditional expectation as an integral on $\mathbb{R}$ with respect to a regular conditional probability.
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