On the surjectivity of the conditional expectation given a real random variable
1 : Ecole des Ponts
CERMICS
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 condi-
tional expectation, f (X) = 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.
| Subject : | : | Presentation |
| Language of text | : | English |
| Date of production, writing | : | 2025-12-23 |
| Topics | : | Flash talks |
| PDF version | : |  PDF version |
