Variational proof of conditional expectations

TL;DR

The paper addresses representing the conditional expectation $\mathbb{E}(X|\mathcal{G})$ as a variational problem in the Hilbert space setting. It constructs the energy functional $J(Y)=\frac{1}{2}\|Y\|_2^2 - \langle X, Y\rangle_2$ on $L^2(\Omega,\mathcal{G},\mathbb{P})$ and shows that $\mathbb{E}(X|\mathcal{G})$ is the unique critical point (Dirichlet principle) and minimum, established via Fréchet-Riesz representation of a continuous linear functional. A density argument extends the existence to all $X\in L^1(\Omega,\mathcal{F},\mathbb{P})$, linking conditional expectation to a variational minimization problem. This work builds on classical projections in $L^2$ and density to provide a variational characterization of conditional expectation, suggesting new analytic tools for martingale theory and related areas. The approach offers a canonical energy-minimization perspective on conditional moments and their extensions.

Abstract

In this paper, we show that the conditional expectation of a random variable with finite second moment given a $σ$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the result to every integrable random variable.

Theorems & Definitions (16)

• Definition 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Definition A.1