Set-valued $$\Phi $$-coherent risk measures for sustainable decision-making

Coherent risk measures, with their rigorous mathematical foundation, are critical for as sessing and managing risks in fields such as sustainability, where uncertainty, incomplete data, and multidimensional risks pose significant challenges. This paper extends the con cept of coherent risk measures by incorporating set-valued probabilities, which represent uncertainty as a family of probability measures, enabling robust risk evaluation under incomplete information. The proposed framework also accommodates multidimensional risks, making it particularly suited for addressing complex sustainability challenges.