= Solution
If $A_0\notin K$, let $p$ be a nearest point of the <closed convex cone> $K$ to $A_0$. Existence follows by minimizing the continuous squared distance on a sufficiently large compact ball. The first-order minimization inequality is $(A_0-p)^\top(y-p)\leq0$ for every $y\in K$. Set $x=p-A_0$. Taking $y=0$ and $y=2p$, both in the cone, gives $x^\top p=0$. Consequently
$$
x^\top y\geq0\quad(y\in K),\qquad x^\top A_0=-\|A_0-p\|^2<0.
$$
The cone's test-weight characterization again yields $x^\top A_1\geq0$ almost surely. This is <separation from a closed convex cone> with the sign needed for
$$
\boxed{x^\top A_0<0,\qquad x^\top A_1\geq0\text{ a.s.}.}
$$
The strict benefit is already available at the initial date, even if the terminal payoff happens to be zero.
Back to article page
