Lagmental Vicfred

Conditional Expectation Is an L2 Projection by Vicfred

Last updated: Sat 27 July 2019

tags: mathematics, martingales, branching processes, optional stopping

For square-integrable variables, conditioning on G is orthogonal projection onto G-measurable variables. A small computation will anchor the general statement before the abstraction takes over.

The data

A process \((M_n,\mathcal F_n)\) is a martingale when \(\mathbf E[M_{n+1}\mid\mathcal F_n]=M_n\). Its baseline \(M_0\) models fair evolution relative to the information currently available.

$$ X\in L^2,\qquad\mathcal G\subseteq\mathcal F $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \mathbf E[(X-\mathbf E[X\mid\mathcal G])Y]=0\quad\forall\,Y\in L^2(\mathcal G) $$

Derivation

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \|X-Y\|_2^2=\|X-\mathbf E[X\mid\mathcal G]\|_2^2+\|\mathbf E[X\mid\mathcal G]-Y\|_2^2 $$

Invariant content

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad X\in L^2,\qquad\mathcal G\subseteq\mathcal F,\\[5pt] \mathsf{C}\;&:\quad \mathbf E[(X-\mathbf E[X\mid\mathcal G])Y]=0\quad\forall\,Y\in L^2(\mathcal G). \end{aligned} $$

Scope

Optional stopping is not valid for every stopping time. Boundedness, integrability, or uniform-integrability conditions prevent hidden mass from escaping at infinity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf E[(X-\mathbf E[X\mid\mathcal G])Y]=0\quad\forall\,Y\in L^2(\mathcal G) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Tue 07 August 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.