math.OC — Optimization and Control

Classical regularity and attainment in portfolio choice with nonhomothetic utility

Contributed by Yuanhong Wei, Zhenyu Sun

We study classical regularity and attainment in terminal-utility portfolio choice with unspanned factors, nonnegative wealth, and exposures that are square-integrable in expectation. A joint estimate for logarithmic marginal utility controls relative risk aversion and factor sensitivity. In several dimensions, a risk-price component uniformly separated from the factor-hedging span gives classical solvability and a unique strong optimal portfolio on every finite horizon. A scalar comparison equation treats correlated single-factor markets. For state-independent $C^4$ utility, the preference assumptions reduce to positive lower and upper bounds on relative risk aversion. In an orthogonal single-factor market, a uniform finite-order condition allows the risk price to vanish and change sign. Subelliptic gradient estimates and a parabolic equation in the factor variable yield a classical value; verification and strict concavity give strong existence for its continuous feedback. In both settings, the optimal dual coefficient has a deterministic bound uniform over initial states and multipliers. Attainment has a separate integrability boundary. For deterministic opportunities with a finite static optimum, the prescribed strategy class attains the value exactly when the optimal payoff lies in $L^2$ under the physical probability. An explicit nonhomothetic utility gives a horizon and wealth classification with smooth finite values and nonattainment. Further results identify wealth-dependent purchasing-power hedges and a strict increase in the dual value under factor-information restrictions.