On the surjectivity of the conditional expectation given a real random variable

In this paper, we investigate the distributions of random couples $(X,Y)$ with $X$ real-valued such that any non-negative integrable random variable $f(X)$ can be represented as a conditional expectation, $f(X)=\mathbb{E}[g(Y)|X]$, for some non-negative measurable function $g$. It turns out that this representation property is related to the smallness of the support of the conditional law of $X$ given $Y$, and in particular fails when this conditional law almost surely has a non-zero absolutely continuous component with respect to the Lebesgue measure. We give a sufficient condition for the representation property and check that it is also necessary under some additional assumptions (for instance when $X$ or $Y$ are discrete). We also exhibit a rather involved example where the representation property holds but the sufficient condition does not. Finally, we discuss a weakened representation property where the non-negativity of $g$ is relaxed. This study is motivated by the calibration of time-discretized path-dependent volatility models to the implied volatility surface.