Conditional Expectation as an L² Projection

Context

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and
$\mathcal G \subseteq \mathcal F$ a sub-$\sigma$-algebra.

We work in the Hilbert space $L^2(\Omega,\mathcal F,\mathbb P)$ with inner product $\langle X,Y\rangle = \mathbb E[XY].$

Define the closed subspace $L^2(\mathcal G) = \{Y \in L^2(\mathcal F) : Y \text{ is } \mathcal G\text{-measurable}\}.$

Theorem (Projection Characterization)

For any $X \in L^2(\mathcal F)$, the conditional expectation
$\mathbb E[X \mid \mathcal G]$ is the unique orthogonal projection of $X$ onto $L^2(\mathcal G)$.

Equivalently: $\mathbb E[X \mid \mathcal G] = \arg\min_{Y \in L^2(\mathcal G)} \mathbb E[(X-Y)^2].$

Orthogonality Property

For all $Z \in L^2(\mathcal G)$, $\mathbb E\big[(X - \mathbb E[X\mid\mathcal G])Z\big] = 0.$

In particular, it is sufficient to check this for
$Z = \mathbf 1_A$, $A \in \mathcal G$.

Proof Sketch

1. $L^2(\mathcal F)$ is a Hilbert space.
2. $L^2(\mathcal G)$ is a closed subspace.
3. Every element of a Hilbert space admits a unique orthogonal projection onto a closed subspace.
4. The projection satisfies the defining property of conditional expectation.

Hence: $\mathbb E[X\mid\mathcal G] = \Pi_{L^2(\mathcal G)}(X).$

Interpretation

• $X$: future payoff or random outcome
• $\mathcal G$: available information
• $\mathbb E[X\mid\mathcal G]$: best mean-square predictor using only $\mathcal G$

This underlies:

• filtering
• least-squares Monte Carlo
• regression-based pricing
• risk-neutral valuation

Finite Partition Example

If $\mathcal G = \sigma(A_1,\dots,A_n)$, $\mathbb E[X\mid\mathcal G] = \sum_{i=1}^n \mathbb E[X\mid A_i]\mathbf 1_{A_i}.$

The projection is piecewise constant on the partition.

Quant Finance Example

Let $X=(S_T-K)^+$ and $\mathcal G=\sigma(S_t)$.