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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 1 / b / Solution

A state-price density is a positive random variable or process that converts future payoffs into present values by expectation. In a one-period zero-cost return model $X$, its normalization is commonly $\mathbb E\rho=1$ and $\mathbb E[\rho X]=0$.


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