M.3.4 Conditional Expectation

Basic Prerequisites: 02 · Probability Spaces & Filtrations and 03 · Distributions & Expectation.

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1. Intuition & Practical Objective

Conditional expectation answers: “given everything I know at time $t$, what is my best guess for the future value $X$?” It is not a number - it is a random variable $\mathbb{E}[X\mid {\mathcal{F}}_t]$ that is measurable w.r.t. the available information ${\mathcal{F}}_t$. On a finite (coin-toss) space it reduces to averaging over the atoms of ${\mathcal{F}}_t$: for each piece of history, average $X$ over all continuations sharing that history (Shreve I §2.3). The measure-theoretic definition replaces this with partial averaging so it works even when the conditioning event has probability zero.

The practical objective: understand the four faces - (1) partial averaging as the defining property; (2) the five properties (linearity, taking-out-what-is-known, tower, independence, conditional Jensen) that make it the pricing tool; (3) the conditional density / bivariate-normal closed form $\mathbb{E}[X\mid Y]=\rho ({\sigma }_1/{\sigma }_2)Y$; (4) the fact that it is the $L^2$ orthogonal projection onto ${\mathcal{F}}_t$ - the exact reason it is the regression function $\mathbb{E}[Y\mid X=x]$ of ESL Ch 2.

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2. Mathematical Ground Truth & Derivations

2.1 Partial averaging definition (Shreve II Def 2.3.1; Shreve I §2.3)

Given sub-$\sigma$-algebra $\mathcal{G}\subseteq \mathcal{F}$, $\mathbb{E}[X\mid \mathcal{G}]$ is the unique (up to a.s.) random variable that is $\mathcal{G}$-measurable and satisfies

$${\int }_A\mathbb{E}[X\mid \mathcal{G}]d\mathbb{P}={\int }_{A}Xd\mathbb{P}\forall A\in \mathcal{G}.$$

Existence is a corollary of the Radon–Nikodym theorem (Shreve I §9.5). On a finite space with atoms, $\mathbb{E}[X\mid {\mathcal{F}}_k](\omega )=$ average of $X$ over the atom containing $\omega$.

2.2 The five properties (Shreve II Thm 2.3.2, eqs 2.3.18–2.3.22)

• (i) Linearity $\mathbb{E}[c_{1}X+c_{2}Y\mid \mathcal{G}]=c_1\mathbb{E}[X\mid \mathcal{G}]+c_2\mathbb{E}[Y\mid \mathcal{G}]$.
• (ii) Taking out what is known - $X$ $\mathcal{G}$-measurable $\Rightarrow$ $\mathbb{E}[XY\mid \mathcal{G}]=X\mathbb{E}[Y\mid \mathcal{G}]$.
• (iii) Tower - $\mathcal{H}\subseteq \mathcal{G}$ $\Rightarrow$ $\mathbb{E}[\mathbb{E}[X\mid \mathcal{G}]\mid \mathcal{H}]=\mathbb{E}[X\mid \mathcal{H}]$.
• (iv) Independence - $X$ independent of $\mathcal{G}$ $\Rightarrow$ $\mathbb{E}[X\mid \mathcal{G}]=\mathbb{E}X$.
• (v) Conditional Jensen - $\phi$ convex $\Rightarrow$ $\mathbb{E}[\phi (X)\mid \mathcal{G}]\ge \phi (\mathbb{E}[X\mid \mathcal{G}])$.

Property (ii) is what lets you pull a known stock price out of a conditional pricing formula; the tower property is what makes backward induction in trees work; (iv) is how the Independence Lemma yields the Markov property of Brownian motion (Shreve II Lemma 2.3.4, §3.5).

2.3 Conditional densities & the bivariate normal (Shreve I Ch 11)

With joint density $f_{X,Y}$ and marginal $f_Y$, the conditional density is $f_{X\mid Y}(x\mid y)=f_{X,Y}(x,y)/f_Y(y)$ and $\mathbb{E}[h(X)\mid Y]=g(Y)$ with $g(y)=\int h(x)f_{X\mid Y}(x\mid y)dx$. For $(X,Y)$ bivariate normal (Shreve I Ex 11.1):

$$X\mid Y=y\sim N(\rho \frac{{\sigma }_1}{{\sigma }_2}y,(1-{\rho }^2){\sigma }_1^2)\Rightarrow \mathbb{E}[X\mid Y]=\rho \frac{{\sigma }_1}{{\sigma }_2}Y.$$

This is the best linear/unbiased square-error estimator of $X$ from $Y$ (Shreve I §11, verified p-129/130).

2.4 $L^2$ projection & the regression function (Shreve II Def 2.3.1; ESL Ch 2)

$\mathbb{E}[X\mid \mathcal{G}]$ is the orthogonal projection of $X$ onto the subspace of $\mathcal{G}$-measurable square-integrable variables: the residual $X-\mathbb{E}[X\mid \mathcal{G}]$ is orthogonal to every $\mathcal{G}$-measurable $V$, $\mathbb{E}[(X-\mathbb{E}[X\mid \mathcal{G}])\cdot V]=0$. This is precisely why the optimal predictor in the mean-square sense is the conditional expectation $f(x)=\mathbb{E}[Y\mid X=x]$ (ESL eq. 2.13), and why the regression MSE decomposes as $\mathbb{E}[\mathrm{Var}(Y\mid X)]+\mathrm{Var}(\mathbb{E}[Y\mid X])$ - the first term is the irreducible noise, the second the reducible part explained by $X$ (ESL eq. 2.46).

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3. Computational Implementation - verify partial-averaging/tower and the normal conditional mean

Simulate a bivariate normal, bin by $X$, and check that the empirical conditional mean tracks $\rho ({\sigma }_1/{\sigma }_2)X$; verify that the $E[Y\mid X]$ predictor beats the constant predictor. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. "$\mathbb{E}[X\mid \{S_t=100\}]$" is undefined by the schoolbook ratio. $\mathbb{P}(A\cap B)/\mathbb{P}(B)$ divides by zero for a zero-probability conditioning event; only partial averaging over the whole $\sigma$-algebra survives (Shreve II §2.3; Shreve I §9.5 existence via Radon–Nikodym).
2. Treating conditional expectation as a number, not a random variable. $\mathbb{E}[X\mid {\mathcal{F}}_t]$ is a function of the path (measurable w.r.t. ${\mathcal{F}}_t$), so it varies across atoms. Writing it as a scalar loses the information-dependence that makes it a martingale’s engine.
3. Believing “best linear” = “best.” $\rho ({\sigma }_1/{\sigma }_2)Y$ is the best among linear (and normal) estimators; for non-normal, non-linear dependencies a different function of $Y$ can beat it. Optimality requires the full conditional expectation (ESL eq. 2.13).
4. Tower property misdirection. $\mathbb{E}[\mathbb{E}[X\mid \mathcal{G}]\mid \mathcal{H}]$ requires $\mathcal{H}\subseteq \mathcal{G}$; conditioning in the wrong order or on a finer $\sigma$-algebra first quietly breaks backward induction.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• ESL (Hastie, Tibshirani, Friedman)
• Williams, Probability with Martingales

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6. Connected Graph Bridges

• Back: 03 · Distributions & Expectation · Index Hub
• Forward: 05 · Martingales (martingale property is a conditional expectation)
• Applications: Black–Scholes–Merton (risk-neutral conditional valuation) · Econometrics & Time Series (conditional mean/volatility as $\mathbb{E}[\cdot \mid {\mathcal{F}}_{t-1}]$)