Existence of an optimal portfolio for SAHARA utilities with a in (0,1] on bounded closed domains

Determine whether the bounded-domain expected utility maximization problem max_{x ∈ D} E[U(W(x))] admits a solution for any bounded closed set D ⊂ R^d when the utility function U is a SAHARA utility with risk-tolerance T(w) = sqrt(b^2 − (w − d)^2) (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 authors derive existence of optimizers under Assumption 1 and present results for convex domains, but they note that SAHARA utilities with a ∈ (0,1] are unbounded and fall outside Assumption 1.

They explicitly state that it is unclear whether the constrained problem (5) on bounded closed domains has a solution in this SAHARA parameter regime and suggest that existence may depend on the mixing distribution Z.

Establishing existence (or non-existence) in this setting would extend their well-posedness results to important classes of utilities not covered by Assumption 1.