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Classical regularity and attainment in portfolio choice with nonhomothetic utility

Primarily human-written textHuman understanding: all partsmath.OC — Optimization and Controlmath.AP — Analysis of PDEsmath.PR — Probability

Contributed by Yuanhong Wei ↗, Zhenyu Sun ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

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 C4C^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 L2L^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.

Provenance statement

ChatGPT was used for proofreading and language editing. The authors retain responsibility for the content of the manuscript. Exact symbolic calculations accompanying the proofs were performed using SymPy.

Tools used

OpenAI
ChatGPT
SymPy
SymPyVersion 1.14.0

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 20, 2026