Conditional Jensen's Inequality

Theorem

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

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

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

Let $f : \R \to \R$ be a convex function such that $\map f X$ is integrable.

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

Then:

$\map f {\expect {X \mid \GG} } \le \expect {\map f X \mid \GG}$

$\ds \map f x = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a x + b}$

so:

$\ds \map f X = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a X + b}$

So for each $\tuple {a, b} \in \SS$, we have:

$a X + b \le \map f X$

$a \expect {X \mid \GG} + b \le \expect {\map f X \mid \GG}$ almost surely

for each $\tuple {a, b} \in \SS$.

For each $\tuple {a, b} \in \SS$, let $A_{\tuple {a, b} }$ be the set of sample points $\omega \in \Omega$ such that this inequality fails.

Then for:

$\ds \omega \in \Omega \setminus \bigcup_{\tuple {a, b} \mathop \in \SS} A_{\tuple {a, b} }$

we have:

$a \map {\expect {X \mid \GG} } \omega + b \le \map {\expect {\map f X \mid \GG} } \omega$ for all $\tuple {a, b} \in \SS$

$\ds \map \Pr {\bigcup_{\tuple {a, b} \mathop \in \SS} A_{\tuple {a, b} } } = 0$

So we have:

$a \expect {X \mid \GG} + b \le \expect {\map f X \mid \GG}$ for all $\tuple {a, b} \in \SS$ almost surely.

We therefore have:

$\ds \sup_{\tuple {a, b} \mathop \in \SS} \paren {a \expect {X \mid \GG} + b} \le \expect {\map f X \mid \GG}$ almost surely.

that is:

$\map f {\expect {X \mid \GG} } \le \expect {\map f X \mid \GG}$ almost surely.

$\blacksquare$

1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list