条件期望的几何意义

设\((\Omega,\mathscr{F},\mathbb{P})\)是概率空间, \(\mathscr{G}\subset\mathscr{F}\)是子\(\sigma\)-域. 考虑随机变量空间(在这里考虑的是二次可积空间, 是为了引入范数和内积. ) \[ \begin{aligned} &\mathscr{L}^2(\mathscr{F})=\{X\in\mathscr{L}^2(\Omega):\text{$X$是$\mathscr{F}$-可测的}\}, \\ &\mathscr{L}^2(\mathscr{G})=\{X\in\mathscr{L}^2(\Omega):\text{$X$是$\mathscr{G}$-可测的}\}. \end{aligned} \] 若随机变量\(X\in\mathscr{L}^2(\mathscr{G})\), 也即\(X\)是\(\mathscr{G}\)-可测的, 根据\(\mathscr{G}\subset\mathscr{F}\), 知 \[ X^{-1}(\mathscr{B}_{\mathbb{R}})\subset\mathscr{G}\subset\mathscr{F}, \] 从而\(X\)一定是\(\mathscr{F}\)-可测的, 这说明了\(\mathscr{L}^2(\mathscr{G})\subset\mathscr{L}^2(\mathscr{F})\). 在空间\(\mathscr{L}^2(\mathscr{F})\)上, 赋予内积 \[ (X, Y):=\left(\int_{\Omega}XY\mathrm{d}\mathbb{P}\right)^{\frac{1}{2}}=\sqrt{\mathbb{E}(XY)},\quad\forall X, Y\in\mathscr{L}^2(\mathscr{F}), \] 则\(\left(\mathscr{L}^2(\mathscr{F}),(\cdot,\cdot)\right)\)为Hilbert空间; 再考虑内积\((\cdot,\cdot)\)诱导的范数 \[ \|X\|:=\sqrt{(X, X)}=\sqrt{\mathbb{E}(X^2)}, \quad\forall X\in\mathscr{L}^2(\mathscr{F}). \] 在\(\mathscr{L}^2(\mathscr{G})\)上, 也赋予相同的内积和范数.

在上面的记号的基础上, 考虑随机变量\(X\in\mathscr{L}^2(\mathscr{F})\), 并记\(X\)关于\(\mathscr{G}\)的条件期望为\(\mathbb{E}(X|\mathscr{G})\). 根据定义知\(\mathbb{E}(X|\mathscr{G})\in\mathscr{L}^2(\mathscr{G})\). 在这里, 我们来说明: \(\mathbb{E}(X|\mathscr{G})\)实质上是\(X\)在空间\(\mathscr{L}^2(\mathscr{G})\)上的投影, 从而条件期望\(\mathbb{E}(\cdot|\mathscr{G})\)实质上是从空间\(\mathscr{L}^2(\mathscr{F})\)到空间\(\mathscr{L}^2(\mathscr{G})\)的投影.

命题1 设\(X\in\mathscr{L}^2(\mathscr{F})\), \(\mathscr{G}\subset\mathscr{F}\)是子\(\sigma\)-域, 则\(\mathbb{E}(X|\mathscr{G})\in\mathscr{L}^2(\mathscr{G})\).

证明 根据定义知\(\mathbb{E}(X|\mathscr{G})\)是\(\mathscr{G}\)-可测的. 又根据\(X\in\mathscr{L}^2(\mathscr{F})\), 应用条件期望的Jensen不等式得 \[ \mathbb{E}\left[\mathbb{E}(X|\mathscr{G})^2\right]\le\mathbb{E}\left[\mathbb{E}(X^2|\mathscr{G})\right]=\mathbb{E} X^2