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We introduce the concept of a risk form, which is a real functional of twoarguments: a measurable function on a Polish space and a measure on that space.We generalize the duality theory and the Kusuoka representation to thissetting. For a risk form acting on a product of Polish spaces, we definemarginal and conditional forms and we prove a disintegration formula, whichrepresents a risk form as a composition of its marginal and conditional forms.We apply the proposed approach to two-stage stochastic programming problemswith partial information and decision-dependent observation distribution.

Risk Forms: Representation, Disintegration, and Application to Partially Observable Two-Stage Systems