Definition

Let $(\Omega,\mathcal F,P)$ be a probability space, let $X$ be integrable, and let $\mathcal G\subseteq \mathcal F$ be a sub-$\sigma$-algebra. The conditional expectation $\mathbb E[X\mid \mathcal G]$ is the random variable $Y$ such that:

  1. $Y$ is $\mathcal G$-measurable.
  2. For every $G\in\mathcal G$,
$$ \int_G Y\,dP=\int_G X\,dP. $$

Equivalently,

$$ \mathbb E[Y\mathbf 1_G]=\mathbb E[X\mathbf 1_G]. $$

Core rules

Linearity

$$ \mathbb E[aX+bY\mid \mathcal G]

a\mathbb E[X\mid \mathcal G]+b\mathbb E[Y\mid \mathcal G]. $$

Taking out what is known

If $Z$ is $\mathcal G$-measurable and $ZX$ is integrable, then

$$ \mathbb E[ZX\mid \mathcal G]

Z\mathbb E[X\mid \mathcal G]. $$

Independence

If $X$ is independent of $\mathcal G$, then

$$ \mathbb E[X\mid \mathcal G]=\mathbb E[X]. $$

Tower property

If $\mathcal H\subseteq \mathcal G\subseteq \mathcal F$, then

$$ \mathbb E[\mathbb E[X\mid \mathcal G]\mid \mathcal H]

\mathbb E[X\mid \mathcal H]. $$

Also,

$$ \mathbb E[\mathbb E[X\mid \mathcal H]\mid \mathcal G]

\mathbb E[X\mid \mathcal H]. $$

Monotonicity

If $X\ge Y$ almost surely, then

$$ \mathbb E[X\mid \mathcal G]\ge \mathbb E[Y\mid \mathcal G]. $$

Conditional Jensen inequality

If $\phi$ is convex and $\phi(X)$ is integrable, then

$$ \phi(\mathbb E[X\mid \mathcal G]) \le \mathbb E[\phi(X)\mid \mathcal G]. $$

Common martingale uses

  • Tower property is used to show
$$ \mathbb E[X_n\mid \mathcal F_m]=X_m \qquad (mfor a martingale.

  • Conditional Jensen gives a standard route from martingale to submartingale. For example, if $(X_n)$ is a martingale, then

$$ X_n^2

\bigl(\mathbb E[X_{n+1}\mid \mathcal F_n]\bigr)^2 \le \mathbb E[X_{n+1}^2\mid \mathcal F_n], $$

so $(X_n^2)$ is a submartingale.