Gaussian two-fund theorem

For nondegenerate Gaussian terminal prices, a finite maximizer of a differentiable increasing concave non-affine utility, subject to a linear budget, lies in the span of $V^{-1}s$ and $V^{-1}\mu$. Gaussian integration by parts givesThe budget first-order condition makes this a multiple of $s$. For a nonzero portfolio, non-affineness and the full Gaussian support imply $\mathbb{E}{U}^{\prime \prime }<0$, so rearrangement gives the span assertion. Affine utility is a genuine exception: when means are proportional to prices, every budget-feasible portfolio is optimal, including portfolios outside that span.