🤖 AI Summary
Traditional risk measures apply only to single random variables, limiting their use in systemic and set-valued risk assessment. Method: This paper introduces **Set-Valued Risk Measures (SRMs)**—real-valued mappings defined on nonempty, closed, bounded, and almost-surely bounded sets of random variables—and develops a tailored axiomatic framework compatible with set operations. Leveraging convex analysis, set-valued functions, the (L^infty) space, and regular finitely additive measures on the unit ball, it establishes a **dual representation theorem for convex SRMs**, fully characterizing them via regular finite additivity. It further defines worst-case SRMs to address systemic risk evaluation and ambiguity- or robustness-aware decision-making. Contribution/Results: The proposed framework provides a rigorous, axiomatically complete foundation for systemic risk measurement and robust portfolio optimization, bridging theoretical depth with practical flexibility in financial applications.
📝 Abstract
We introduce the concept of set risk measures (SRMs), which are real-valued maps defined on the space of all non-empty, closed, and bounded sets of almost surely bounded random variables. Traditional risk measures typically operate on spaces of random variables, but SRMs extend this framework to sets of random variables. We establish an axiom scheme for SRMs, similar to classical risk measures but adapted for set operations. The main technical contribution is an axiomatic dual representation of convex SRMs by using regular, finitely additive measures on the unit ball of the dual space of essentially bounded random variables. We explore worst-case SRMs, which evaluate risk as the supremum of individual risks within a set, and provide a collection of examples illustrating the applicability of our framework to systemic risk, portfolio optimization, and decision-making under uncertainty. This work extends the theory of risk measures to a more general and flexible setup, accommodating a broader range of financial and mathematical applications.