🤖 AI Summary
This study addresses utility maximization for an investor facing an exogenous contingent claim with known marginal distributions but ambiguous dependence structure. Introducing an α-robust criterion that continuously interpolates between worst- and best-case scenarios, the dynamic stochastic control problem is transformed into a static concave quantile optimization over a convex set. The optimal quantile function is derived via variational methods. Innovatively integrating the α-robust framework with quantile optimization, the analysis leverages the rearrangement inequality and comonotonicity theory to establish distributional invariance of risk measures, thereby naturally accommodating risk constraints such as Value-at-Risk (VaR) and Expected Shortfall (ES). The solution yields a numerically tractable two-dimensional first-order ordinary differential equation system, elucidating the joint impact of ambiguity aversion, market conditions, and claim characteristics on the optimal payoff structure.
📝 Abstract
This paper studies an $α$-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 $α$-robust criterion interpolates between worst-case ($α=0$) and best-case ($α=1$) evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the $α$-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.