(Perpetual American put paying dividends). Consider a perpetual American put on a geometric Brownian motion asset price paying dividends at a constant rate $a>0$. The differential of this asset is
$$
d S(t)=(r-a) S(t) d t+\sigma S(t) d \widetilde{W}(t),
$$
where $\widetilde{W}(t)$ is a Brownian motion under a risk-neutral measure $\widetilde{\mathbb{P}}$. (Equation (8.8.9) can be obtained by computing the differential in (5.5.8).)
(i) Suppose we adopt the strategy of exercising the put the first time the asset price is at or below $L$. What is the risk-neutral expected discounted payoff of this strategy? Write this as a function $v_L(x)$ of the initial asset price $x$. (Hint: Define the positive constant
$$
\gamma=\frac{1}{\sigma^2}\left(r-a-\frac{1}{2} \sigma^2\right)+\frac{1}{\sigma} \sqrt{\frac{1}{\sigma^2}\left(r-a-\frac{1}{2} \sigma^2\right)^2+2 r}
$$