= Solution
The same pricing relation gives $x^\top A_0\geq0$ whenever the terminal payment is nonnegative. Therefore negative initial cost also contradicts a strictly positive <state-price density>, even if the terminal payment is identically zero.
This second form is needed for directions invisible in the terminal payments. Let $L$ be the <linear span> of the essential values of $A_1$, equivalently
$$
L=\{u\in\mathbb R^r:u^\top A_1=0\text{ almost surely}\}^{\perp}.
$$
If $A_0\notin L$, let $v$ be its nonzero <orthogonal projection> on $L^\perp$ and set $x=-v$. Then $x^\top A_1=0$ <almost surely> while $x^\top A_0=-\|v\|^2<0$. Hence the second form exists in this case. Part (b) handles all remaining cases.
Back to article page
