Robust Utility Maximization with Intractable Claims under Distributional Ambiguity: A Random Distributionally Robust Optimization Approach

This paper studies a robust utility maximization problem for intractable claims under distributional ambiguity, where the distribution of the claim cannot be inferred from market information and its dependence with tradable assets is largely unknown. We extend the existing framework for intractable claims in two directions. First, we allow the marginal distribution of the claim to vary within a $\phi$-divergence ambiguity set, capturing statistical uncertainty in its estimation. Second, we consider a general (possibly non-additive) bivariate utility function, which enables more flexible interactions between the decision and the claim beyond the classical additive specification. To analyze this problem, we adopt a random distributionally robust optimization (RDRO) formulation, which lifts the optimization to the space of joint distributions and provides a convenient representation of the coupling between the decision and the uncertain claim. We establish the existence of optimal decisions using tools from optimal transport and develop a Legendre-Fenchel duality framework that links the constrained and penalized formulations, leading to uniqueness results and tractable reformulations. Finally, we propose a numerical algorithm based on unbalanced optimal transport scaling combined with projected gradient methods, and illustrate the relationship between the parameters in the constrained and penalized formulations.