Rule for Extracting Random Variable from Conditional Expectation of Product

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Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

Let $X$ and $Y$ be integrable random variables such that:

$X Y$ is integrable

and:

$Y$ is $\GG$-measurable.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Let $\expect {X Y \mid \GG}$ be a version of the conditional expectation of $X Y$ given $\GG$.


Then:

$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere.


Proof

Let $X$ and $Y$ be non-negative random variables.

We show first that the statement holds if $Y = \chi_A$ for some $A \in \GG$.

We show that $Y \expect {X \mid \GG}$ is a version of the conditional expectation of $X Y$ given $\GG$.

We will then obtain the demand from Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra.

Then, for any $G \in \GG$ we have:

\(\ds \expect {Y \expect {X \mid \GG} \cdot \chi_G}\) \(=\) \(\ds \expect {\expect {X \mid \GG} \cdot \chi_{A \cap G} }\) Characteristic Function of Intersection
\(\ds \) \(=\) \(\ds \expect {X \cdot \chi_{A \cap G} }\) Definition of Conditional Expectation on Sigma-Algebra
\(\ds \) \(=\) \(\ds \expect {X \cdot \chi_A \cdot \chi_G}\) Characteristic Function of Intersection
\(\ds \) \(=\) \(\ds \expect {X Y \cdot \chi_G}\)

So we have:

$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere

for all $Y$ of the form $\chi_A$ for $A \in \GG$.


Now let $Y$ be a positive simple real-valued random variable that is $\GG$-measurable.

From Simple Function has Standard Representation, there exists:

a finite sequence $a_1, \ldots, a_n$ of real numbers
a partition $A_1, A_2, \ldots, A_n$ of $\GG$-measurable sets

with:

$\ds Y = \ds \sum_{j \mathop = 1}^n a_j \chi_{A_j}$

We then have:

\(\ds \expect {X Y \mid \GG}\) \(=\) \(\ds \expect {X \paren {\sum_{j \mathop = 1}^n a_j \chi_{A_j} } \mid \GG}\)
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^n a_j \expect {X \chi_{A_j} \mid \GG}\) Conditional Expectation is Linear
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^n a_j \chi_{A_j} \expect {X \mid \GG}\) applying the theorem for characteristic functions
\(\ds \) \(=\) \(\ds \expect {X \mid \GG} \sum_{j \mathop = 1}^n a_j \chi_{A_j}\)
\(\ds \) \(=\) \(\ds Y \expect {X \mid \GG}\)


Now let $Y$ be a general non-negative random variable.

Similarly, there exists an increasing sequence $\sequence {Y_n}_{n \mathop \in \N}$ of positive simple random variables such that:

$\ds \map Y \omega = \lim_{n \mathop \to \infty} \map {Y_n} \omega$

for $\omega \in \Omega$.

Then $\sequence {X Y_n}_{n \mathop \in \N}$ is an increasing sequence of random variables.

So, by the Conditional Monotone Convergence Theorem, we have:

$\ds \lim_{n \mathop \to \infty} \expect {X Y_n \mid \GG} = \expect {X Y \mid \GG}$ almost everywhere.

We also see that:

$\ds \lim_{n \mathop \to \infty} Y_n \expect {X \mid \GG} = Y \expect {X \mid \GG}$

Hence we have:

$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere

when $X$ and $Y$ are non-negative.


Now, suppose that $X$ and $Y$ are general integrable random variables.

Then we have:

\(\ds \expect {X Y \mid \GG}\) \(=\) \(\ds \expect {\paren {X^+ - X^-} \paren {Y^+ - Y^-} \mid \GG}\)
\(\ds \) \(=\) \(\ds \expect {X^+ Y^+ - X^- Y^+ - X^+ Y^- + X^- Y^- \mid \GG}\)
\(\ds \) \(=\) \(\ds \expect {X^+ Y^+ \mid \GG} - \expect {X^- Y^+ \mid \GG} - \expect {X^+ Y^- \mid \GG} + \expect {X^- Y^- \mid \GG}\) Conditional Expectation is Linear

From Function Measurable iff Positive and Negative Parts Measurable, we have that:

$Y^+$ and $Y^-$ are $\GG$-measurable.

Then we have, since the theorem is known to hold for non-negative random variables:

\(\ds \expect {X^+ Y^+ \mid \GG} - \expect {X^- Y^+ \mid \GG} - \expect {X^+ Y^- \mid \GG} + \expect {X^- Y^- \mid \GG}\) \(=\) \(\ds Y^+ \expect {X^+ \mid \GG} - Y^+ \expect {X^- \mid \GG} - \paren {Y^- \expect {X^+ \mid \GG} - Y^- \expect {X^- \mid \GG} }\)
\(\ds \) \(=\) \(\ds Y^+ \expect {X^+ - X^- \mid \GG} - Y^- \expect {X^+ - X^- \mid \GG}\) Conditional Expectation is Linear
\(\ds \) \(=\) \(\ds \expect {X \mid \GG} \paren {Y^+ - Y^-}\)
\(\ds \) \(=\) \(\ds Y \expect {X \mid \GG}\)

$\blacksquare$


Also known as

Applying this rule is often known as taking out what is known.


Sources

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