🤖 AI Summary
This study addresses risk measurement for financial positions in spaces of Lipschitz functions lacking constant terms and Banach lattice structure. Taking Lipschitz functions vanishing at a reference state as the natural domain, it pioneers a dual representation framework by integrating Lipschitz-free spaces with optimal transport theory. The approach overcomes the absence of cash-additivity through an additive mechanism based on benchmark deviations. This methodology not only yields a unified risk measurement model applicable to complex financial settings—such as path-dependent payoffs, temporal cash flows, and network structures—but also extends to scenarios involving model uncertainty. Within this broader context, the paper successfully derives dual representations for both convex and coherent risk measures, substantially expanding the applicability of classical risk measurement theory.
📝 Abstract
This paper develops a theory of monetary risk measures on metric state spaces. We propose the space of Lipschitz functions vanishing at a reference state as a natural domain for financial positions. The associated Lipschitz-free space provides its canonical predual, linking anchored Lipschitz payoffs to transport-based dual variables interpreted as redistributions of mass around the benchmark. Since the domain lacks constants and need not be a Banach lattice under the Lipschitz norm, standard cash-additive methods do not apply directly. We address this by using additivity along benchmark-deviation instruments and derive dual representations for convex and coherent risk measures. The framework covers temporal cash flows, path-dependent payoffs, network risk, and model uncertainty.