Solution
= Solution
Let $(h_k)$ be a bounded minimizing sequence. A subsequence converges to some $h_*$, and continuity gives $F(h_*)=f$. Since $F$ is smooth and $h_*$ is an unconstrained minimizer,
$$
0=\nabla F(h_*)
=-\mathbb E[Xe^{-h_*\cdot X}\zeta].
$$
Define
$$
\rho=\frac{e^{-h_*\cdot X}\zeta}{F(h_*)}.
$$
Then $\rho>0$, $\mathbb E\rho=1$, and
$$
\mathbb E[\rho X]
=-\frac{\nabla F(h_*)}{F(h_*)}=0.
$$
Thus the normalized exponential tilt is the required <state-price density>.