Consider two perpetual American puts on the geometric Brownian motion (8.3.1). Suppose the puts have different strike prices, $K_1$ and $K_2$, where $0<K_1<K_2$. Let $v_1(x)$ and $v_2(x)$ denote their respective prices, as determined in Section 8.3.2. Show that $v_2(x)$ satisfies the first two linear complementarity conditions,
$$
\begin{aligned}
v_2(x) & \geq\left(K_1-x\right)^{+} \text {for all } x \geq 0, \\
r v_2(x)-r x v_2^{\prime}(x)-\frac{1}{2} \sigma^2 x^2 v_2^{\prime \prime}(x) & \geq 0 \text { for all } x \geq 0,
\end{aligned}
$$
for the perpetual American put price with strike $K_1$ but that $v_2(x)$ does not satisfy the third linear complementarity condition:
for each $x \geq 0$, equality holds in either (8.8.1) or (8.8.2) or both.