Set risk measures
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.