1.7 Conditional expectation

1.7.3 regular conditional probability

Definition 1.42 Regular conditional probability

• $(\Omega, \mathcal{F}, P)$,
  • probability sp.
• $\mathcal{G} \subseteq \mathcal{F}$,
  • sub $\sigma$-algebra
• $p(\omega, A) \ (\omega \in \Omega, A \in \mathcal{A}) \in [0, 1]$

\(\{p(\omega, A) \}_{\omega \in \Omega, A \in \mathcal{F}}\) is said to be regular conditional probability given $\mathcal{G}$ if it satisfies

• (1) For all $\omega \in \Omega$, a map $p(\omega, \cdot): \mathcal{F} \rightarrow [0, 1]$ is probability measure over $(\Omega, \mathcal{F})$,
• (2) For all $A \in \mathcal{A}$, a map $p(\cdot, A): \Omega \rightarrow [0, 1]$ is $\mathcal{G}$ measurable function,
• (3)

\[\forall A \in \mathcal{F}, \ \forall B \in \mathcal{B}, \ P(A \cap B) = \int_{B} p(\omega, B) \ P(d \omega) .\]

Regular conditional probability \(\{p(\omega, A)\}\) is said to be unique if for all \(\{p^{\prime}(\omega, A) \}\) satisfying (1), (2) and (3), there exists $P$-null set $N \in \mathcal{G}$ such that

\[\forall \omega \notin N, \ \forall A \in \mathcal{F}, \ p(\omega, A) = p(\omega^{\prime}, A)\]

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Definiiton Condition (S)

• $(\Omega, \mathcal{F})$,
  • measurable sp.

$(\Omega, \mathcal{F})$ is said to satisfy condition (S) if there exists distance $d$ such that

• $(\Omega, d)$ is complete separable distance space,
• $\mathcal{F}$ is borrel $\sigma$ algebra of $(\Omega, d)$.

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Remark

• $X: \Omega \rightarrow \mathbb{R}$
  • integrable r.v.

\[\mathrm{E} \left[ X \mid \mathcal{G} \right](\omega) = \int_{\Omega} X(\omega^{\prime}) \ p(\omega, d \omega^{\prime}\]

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Theorem 1.43

• $(\Omega, \mathcal{F})$,
  • satisfies condition (S)
• $P$
  • probability measure over $(\Omega, \mathcal{F})$,
• $\mathcal{G} \subseteq \mathcal{F}$,
  • sub $\sigma$-algebra

Then there uniquely exists regular probibility measure \(\{p(\omega, A)\}_{\omega \in \Omega, A \in \mathcal{A}}\) given $\mathcal{G}$.

proof

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Definition 1.44 Regular conditional distribution

• $(\Omega, \mathcal{F})$,
  • measurable sp.
• $(\mathcal{X}, \mathcal{A})$,
  • measurable sp.
• $(\mathcal{T}, \mathcal{B})$,
  • measurable sp.
• $X: \Omega \rightarrow \mathcal{X}$,
  • measurable
• $T: \Omega \rightarrow \mathcal{T}$,
  • measurable
• $P: \Omega \rightarrow [0, 1]$
  • probability measure over $(\Omega, \mathcal{F})$
• $p(t, A) \in [0, 1]\ (t \in \mathcal{T}, A \in \mathcal{A})$

\(\{p(t, A)\}_{t \in \mathcal{T}, A\in \mathcal{A}}\) is said to be regular conditional distribution of $X$ given $T = t$ if it satisfies

• (1) For all $t \in \mathcal{T}$, a map $p(t, \cdot): \mathcal{A} \rightarrow [0, 1]$ is probability measure over $(\mathcal{X}, \mathcal{A})$,
• (2) For all $A \in \mathcal{A}$, a map $p(\cdot, A): \mathcal{X} \rightarrow [0, 1]$ is $\mathcal{B}$ measurable,
• (3)

\[\forall A \in \mathcal{A}, \ \forall B \in \mathcal{B}, \ P(X \in A, T \in B) = \int_{B} p(t, A) \ P^{T}(dt) .\]

Regular conditional probability \(\{p(t, A)\}\) of $X$ given $T = t$ is said to be unique if for all \(\{p^{\prime}(t, A) \}\) satisfying (1), (2) and (3) there exists $P^{T}$-null set $N \in \mathcal{B}$ such that

\[\forall t \notin N, \ \forall A \in \mathcal{A}, \ p(t, A) = p(t^{\prime}, A) .\]

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Theorem 1.45

• $(\mathcal{X}, \mathcal{A})$,
  • measurable sp.
  • satisfying condition (S)

Then there uniquely exists regular probibility measure \(\{p(\omega, A)\}_{\omega \in \Omega, A \in \mathcal{A}}\) of $X$ given $T = t$.

proof

$\Box$

Remark

• $f: \mathcal{T} \times \mathcal{X} \rightarrow \mathbb{R}$,
  • $\mathcal{B} \times \mathcal{A}$ measurable function
  • $P^{(T, X)}$ integrable

\[\int_{\mathcal{T} \times \mathcal{X}} f(t, x) \ P^{(T, X)}(dt, dx) = \int_{\mathcal{T}} \int_{\mathcal{X}} f(t, x) \ p(t, dx) \ P^{T}(dt) .\]

In particular, integrable real-valued r.v. $X$,

\[\mathrm{E} \left[ X \mid T = t \right] = \int_{\mathbb{R}} x \ p(t, dx) \quad P^{T} \text{-a.s.}\]

Indeed,

\[\begin{eqnarray} \forall B \in \sigma(T), \ \int_{\mathcal{T}} 1_{B}(t) \int_{\mathcal{X}} x \ p(t, dx) \ P^{T}(dt) & = & \int_{\mathcal{T}} \int_{\mathcal{X}} 1_{B}(t) x \ p(t, dx) \ P^{T}(dt) \nonumber \\ & = & \int_{\mathcal{T} \times \mathcal{X}} 1_{B}(t) x \ P^{(T, X)}(dt, dx) \nonumber \\ & = & \int_{\mathcal{T}} 1_{B}(t) \int_{\mathcal{X}} x \ P^{X}(dx) \ P^{T}(dt) \nonumber \\ & = & \int_{\mathcal{T}} 1_{B}(t) \int_{X^{-1}(\mathcal{X})} X(\omega) \ P(d\omega) \ P^{T}(dt) \nonumber \end{eqnarray}\]

Here we use the equation above as $f(t, x) \equiv 1_{B}(t)x$ and Fubini-Tonelli theorem. Then we interpret the range of r.v. as $\mathcal{X} = \mathcal{T} = \mathbb{R}$.

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