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

  • Darinka Dentcheva
  • Andrzej RuszczyńskiEmail author
Full Length Paper Series B


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.


Risk measures Kusuoka representation Risk decomposition Two-stage stochastic programming Partially observable systems 

Mathematics Subject Classification

90C15 90C48 



We thank two anonymous Referees for insightful comments that helped improve our paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

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