$\alpha$-robust utility maximization with intractable claims: A quantile optimization approach

This paper studies an $\alpha$-robust utility maximization problem where an investor faces an intractable claim -- an exogenous contingent claim with known marginal distribution but unspecified dependence structure with financial market returns. The $\alpha$-robust criterion interpolates between worst-case ($\alpha=0$) and best-case ($\alpha=1$) evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the $\alpha$-robust risk measure is law-invariant, depending only on marginal distributions. This transforms the dynamic stochastic control problem into a concave static quantile optimization over a convex domain. We derive optimality conditions via calculus of variations and characterize the optimal quantile as the solution to a two-dimensional first-order ordinary differential equation system, which is a system of variational inequalities with mixed boundary conditions, enabling numerical solution. Our framework naturally accommodates additional risk constraints such as Value-at-Risk and Expected Shortfall. Numerical experiments reveal how ambiguity attitude, market conditions, and claim characteristics interact to shape optimal payoffs.