Consider a filtration $\{\mathcal{F}_t\}_{t\geq 0}$. If the process $\{X_t\}$ is $\mathcal{F}$-measurable, then $\{Y_t\}_{t\geq 0}$ defined as $$Y_t=E\Big[\int_0^\infty e^{-ks}X_sds\vert\mathcal{F}_t\Big]$$ where $k>0$ is a martingale (I think I am right here).
I would like to show that the variance of $Y$ is decreasing with $t$. My intuition is that over time, we learn more about $X$ and what is left to learn has a smaller weight because of the decreasing exponential. But I can't find a way of proving this formally. Any pointers appreciated. Thanks.
Variances in fact increase with $t$. If $s<t$ then $Y_s=E(Y_t|\mathcal F_s)$. By conditional of Jensen's inequality we get $EY_s^{2} \leq EE(Y_t^{2} |\mathcal F_s) =EY_t^{2}$. Hence the second moments are increasing. Now use the fact that $EY_t$ is independent of $t$ to see that variances are also increasing.