Robust Risk Under Evolving Uncertainty: A Wasserstein Counterpart of the Entropic Value-at-Risk

An agent still learning its environment should be cautious while ignorant and bold once confident. The entropic value-at-risk captures this through a robust-optimization identity---a confidence level fixes the radius of a relative-entropy ball of alternative models---but that ball cannot reach catastrophes the nominal deems impossible, precisely what a safe agent must hedge. We instead use an optimal-transport ball and study the coherent risk measure it induces, the Wasserstein entropic value-at-risk. It has a variational dual mirroring the entropic formula (an inverse temperature becomes a transport price), occupies a definite place in the risk hierarchy, and provably accounts for the reachable catastrophes the entropic measure ignores; we verify both dualities numerically. Driving the transport radius by belief entropy then yields a closed-form robust dynamic-programming operator whose caution contracts as the belief sharpens, with a certified safety sandwich and a sharp safety switch.