  In this paper, we study convex risk measures with weak optimal transport
penalties. In a first step, we show that these risk measures allow for an
explicit representation via a nonlinear transform of the loss function. In a
second step, we discuss computational aspects related to the nonlinear
transform as well as approximations of the risk measures using, for example,
neural networks. Our setup comprises a variety of examples, such as classical
optimal transport penalties, parametric families of models, uncertainty on path
spaces, moment constrains, and martingale constraints. In a last step, we show
how to use the theoretical results for the numerical computation of worst-case
losses in an insurance context and no-arbitrage prices of European contingent
claims after quoted maturities in a model-free setting.
