Overview
Measure Theoretic Conditional Expectation
In measure theory, the conditional expectation is defined to be a random variable, where the traditional definition is the value of the measure theoretic random variable on a given datapoint.
The conditional expectation of {% X %} given {% Y %} (labeled {% \mathbb{E}[X|Y] %}) is a {% \sigma(Y) %} -measurable random variable such that
{% \displaystyle \int_A \mathbb{E}[X|Y] dP = \int_A X dP %}
for {% A \in \sigma(Y) %}
Properties of Conditional Expectation
Given sigma algebras {% \mathcal{G} %} and {% \mathcal{H} %}
- Linearity - {% \mathbb{E}[aX + bY | \mathcal{G}] = a\mathbb{E}[X|\mathcal{G}] + b\mathbb{E}[Y|\mathcal{G}] %}
- {% \mathbb{E}[\mathbb{E}[X|\mathcal{G}]] = \mathbb{E}[X] %}
- If {% X %} is {% \mathcal{G} %} measurable, {% \mathbb{E}[X|\mathcal{G}] = X \, a.s. %}
- if {% \sigma(X) %} and {% \mathcal{G} %} are independent, {% \mathbb{E}[X|\mathcal{G}] = \mathbb{E}[X] %}
- Tower Property - {% \mathbb{E}[\mathbb{E}[X|\mathcal{G}]|\mathcal{H}] = \mathbb{E}[X|\mathcal{H}] %}
Filtration
A filtration is a function
{% t \mapsto \mathcal{F}(t) %}
where {% \mathcal{F}(t) %} is a sigma algebra, and
for {% t_2 > t_1 %} we have {% \mathcal{F}(t_1) \subset \mathcal{F}(t_2) %}
Filtrations are meant to capture the passage of time. That is, for a probability space {% \Omega %}, the filtration {% \mathcal{F}(t) %} represents the set of events (sets) for which it is known at time {% t %} whether the event has occurred or not.