Extension, stability and smoothing of risk forms

Abstract Risk forms are real-valued functionals originally defined on the product of two spaces: the space of bounded Borel-measurable functions on a Polish space $$\mathfrak {D}$$ and the space of probability measures on $$\mathfrak {D}$$ . We extend the definition to the fibered set of the space of probability measures on a Polish space $$\mathfrak {D}$$ equipped with an appropriate transportation distance and the associated Borel measurable functions on $$\mathfrak {D}$$ . We establish a Kusuoka representation of law-invariant coherent risk forms, in which the set of mixing measures does not depend on the reference probability measure. Additionally, we obtain quantitative stability of the risk forms with respect to the reference probability measure, which includes Lipschitz-continuity of the dual sets, as well as Lipschitz-continuity of the risk-form value when Lipschitz functions are used. We apply our results to analyze the effect of smoothing the reference probability measure, defined by a convolution with a centered atomless probability measure. Bounds quantifying the difference between the smoothed risk forms and the base risk forms are obtained, providing rigorous theoretical guarantees. The smoothed forms exhibit significantly faster convergence rate that can be leveraged to improve the risk evaluation in numerical methods.