Conditional Riesz Representation

Question

Let $(\Omega,\mathcal{F},\left\{\mathcal{F}_t\right\}_{t\in[0,T]},\mathbb{P})$ be a filtered probability space. Suppose that $S:L^2(\mathcal{F}_T)\longrightarrow L^2(\mathcal{F}_t)$ is a continuous, linear operator where $t\in[0,T]$. Is there a chance to find so ''conditonal Riesz representation'' of $S$, i.e. a random variable $Z_S$ satisfying $$S(X)=\mathbb{E}\left[Z_S X|\mathcal{F}_t\right]$$ for all $X\in L^2(\mathcal{F}_T)$?

Answer 1

As far as I see, we need the assumption $1_A S(X)=S(1_A X)$ for all $X\in L^2_T$ and $A\in \mathcal{F}_t$.

For simpliciness of notation, let $L^2_T:=L^2(\mathcal{F}_T)$. Consider $\bar{S}:L^2_T\rightarrow \mathbb{R}$, defined by $\bar{S}(X):=\mathbb{E}[S(X)]$ for $X\in L^2_T$. Clearly $\bar{S}$ is linear. Also, it is not difficult to verify that it is continuous.

By the classical Riesz representation theorem, one can find $Y\in L^2_T$ such that $\bar{S}(X)=\mathbb{E}[X Y]$ for all $X\in L^2_T$.

Then, we claim that ${S}(X)=\mathbb{E}[X Y|\mathcal{F}_t]$ for all $X\in L^2_T$.

Indeed, given $A\in\mathcal{F}_t$, we have that $\mathbb{E}[1_A X Y]=\bar{S}(1_A X)=\mathbb{E}[S(1_A X)]=\mathbb{E}[1_A S(X)]$.

Then, by definition of conditional expectation one has ${S}(X)=\mathbb{E}[X Y|\mathcal{F}_t]$ for all $X\in L^2_T$.