Existence of an optimal portfolio for SAHARA utilities with a in (0,1] on the full portfolio domain

Determine whether the unconstrained expected utility maximization problem max_{x ∈ R^d} E[U(W(x))] admits a solution when the utility function U is a SAHARA utility with risk-tolerance T(w) = sqrt(b^2 − (w − d)^2) (yielding the forms in equations (51)–(52)) with parameter a ∈ (0,1], and the d-dimensional risky-asset log-return vector X follows a normal mean–variance mixture distribution X = μ + yZ + √Z A N as in equation (1), with wealth W(x) = W0(1 + rf) + W0[x^T(X − 1 rf)] as in equation (4).

Background

The paper studies one-period expected utility maximization with returns modeled by normal mean–variance mixture (NMVM) distributions and provides closed-form characterizations and two-fund separation when utility functions satisfy Assumption 1 (continuous, non-decreasing, bounded above, and lim_{w→−∞} U(w) = −∞).

SAHARA utilities with parameter a > 1 satisfy Assumption 1, so existence and uniqueness are established. However, for SAHARA utilities with a ∈ (0,1], the utility is unbounded above and below and does not satisfy Assumption 1. The authors explicitly state that in this case it remains unclear whether the unconstrained problem (47) has a solution, suggesting that existence may depend on properties of the mixing distribution Z.

Resolving this would clarify the well-posedness of expected utility maximization for widely used SAHARA utilities in NMVM return models beyond the regime covered by Assumption 1.