= Gaussian two-fund theorem
{c}
Let terminal asset prices $S$ have mean $\mu$ and positive definite <covariance matrix> $V$, with a <Gaussian> distribution. Consider a finite optimizer of $\mathbb E[U(\pi^TS)]$ subject to the linear budget $s^T\pi=w$. Assume $U\in C^2(\mathbb R)$ is increasing, concave and non-affine, and its derivatives satisfy the <integrability> needed for differentiation and <Gaussian integration by parts>. Then every optimizer lies in the <linear span> of $V^{-1}s$ and $V^{-1}\mu$.
The first-order condition and Gaussian identity give
$$
\mu\,\mathbb E U'(\pi^TS)+V\pi\,\mathbb E U''(\pi^TS)=\lambda s.
$$
For $\pi\ne0$, the terminal wealth has full Gaussian support. Continuity and non-affine concavity make $U''<0$ on some interval, so $\mathbb E U''(\pi^TS)<0$. Rearranging proves the two-fund representation. The zero portfolio belongs to the same span automatically.
The regularity conditions justify the displayed derivative proof. Affine utility is an exception to its conclusion: if means are proportional to initial prices, every budget-feasible portfolio can be optimal, including portfolios outside this span.
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