Risk forms: representation, disintegration, and application to partially observable two-stage systems

We introduce the concept of a risk form, which is a real functional of two arguments: a measurable function on a Polish space and a measure on that space. We generalize the duality theory and the Kusuoka representation to this setting. For a risk form acting on a product of Polish spaces, 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 stochastic programming problems with partial information and decision-dependent observation distribution.