= Exponential minimization construction of a bounded pricing kernel
{title2=$Y=e^{-h_*\cdot P}$}
For a bounded payoff vector $P$, the condition that $h\cdot p\leq0\leq h\cdot P$ almost surely forces $h=0$ makes $F(h)=\mathbb E e^{-h\cdot P}+h\cdot p$ coercive. At a minimizer $h_*$, differentiation gives $\mathbb E[Pe^{-h_*\cdot P}]=p$. The resulting positive <pricing kernel> is bounded above and away from zero. A direct coercivity proof uses compactness of the unit sphere and separates directions with negative-payoff probability from directions with nonnegative payoffs.
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