Risk Measures on Lipschitz Spaces

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.