Risk Forms: Duality, Disintegration, and Statistical Estimation

We introduce the concept of a risk form, which is a real functional on
the product of two spaces: the space of measurable functions and the
space of measures on a Polish space. We present a dual representation
of risk forms and generalize the classical Kusuoka representation to
this setting. For a risk form acting on a product space, we define
marginal and conditional forms and we prove a disintegration formula,
which represents a risk form as a composition of its marginal and
conditional forms. We apply the proposed approach to two-stage
optimization problems with partial information and decision-dependent
observation distribution. Finally, we discuss statistical estimation
of risk forms and present a central limit formula for a class of forms
defined by nested expectations.