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

TL;DR

This paper introduces risk forms, a new mathematical framework for representing risk in complex systems, generalizing existing theories and applying to two-stage stochastic programming with partial information.

Contribution

It develops the theory of risk forms, including duality, representation, and disintegration, and applies these concepts to advanced stochastic programming problems.

Findings

Established a disintegration formula for risk forms.

Generalized duality and Kusuoka representation for risk forms.

Applied the framework to two-stage stochastic programming with partial information.

Abstract

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.