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

TL;DR

This paper explores when a real-valued random variable allows any non-negative integrable function of it to be represented as a conditional expectation, revealing conditions related to the support of the conditional law and implications for financial modeling.

Contribution

It provides necessary and sufficient conditions for the representation property of conditional expectations involving real random variables, including examples and relaxed conditions.

Findings

Representation property fails with non-zero absolutely continuous component

Sufficient and necessary conditions are identified for the property

Example where the property holds without the sufficient condition

Abstract

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.