Risk Forms: Disintegration, Statistical Estimation, and Subgradient Optimization
[CANCELLED] Program in Applied Mathematics Colloquium
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. Then, we discuss statistical estimation of risk forms and present a central limit formula for a class of forms defined by nested expectations. Finally, we briefly discuss stochastic recursive methods for optimizing risk forms.