Market-dependent well-posedness for utility–risk portfolio selection

Determine, for a fixed pair consisting of an increasing utility functional U: L → [−∞, ∞) and a decreasing risk functional R: L → (−∞, ∞], the set of arbitrage-free one-period markets on L for which the utility–risk portfolio selection problem that maximizes U(π̄ · S̄1) over portfolios π̄ ∈ ℝ1+d subject to the constraints π̄ · S̄0 = w and R(π̄ · S̄1) ≤ Rmax admits an optimizer. Equivalently, provide market-dependent necessary and sufficient conditions characterizing exactly which markets yield well-posedness for the (U, R) problem, beyond the paper’s model-independent sensitivity-to-large-losses criterion.

Background

The paper establishes a model-independent characterization of well-posedness for one-period utility–risk portfolio selection: under mild regularity, well-posedness across all markets holds if and only if either the utility functional or the risk functional is sensitive to large losses. This provides a minimal, general criterion that applies to a wide range of utility and risk functionals, including non-concave S-shaped utilities and non-convex risk measures.

The authors note that, beyond this market-independent result, a market-dependent characterization remains unresolved. They highlight a known special case—mean utility with Expected Shortfall at level α—where well-posedness is ensured for markets admitting an equivalent martingale measure with Radon–Nikodym derivative bounded by 1/α in L∞. A general characterization for arbitrary (U, R) pairs and markets would delineate precisely which market models yield existence of optimizers, providing sharper conditions for practical applications and regulation.