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In Carr's 1998 paper Randomization and the American Put, he sets up the following ODE for the value of an American put with expiration given by the first jump time of a Poisson process with rate $\lambda = 1/T$ (Eq. 10 in link): $$ \frac{\sigma^2}{2}S^2 P^{(1)}_{SS}(S) + rSP^{(1)}_S(S) - rP^{(1)}(S) = \lambda[P^{(1)}(S) - (K - S)^+], \qquad S > \underline{S}_1 \quad (1) $$ subject to the boundary conditions $$ \lim_{S \, \uparrow \infty} P^{(1)}(S) = 0, \qquad \lim_{S \, \downarrow \underline{S}_1} P^{(1)}(S) = K - \underline{S}_1, \qquad \lim_{S \, \downarrow \underline{S}_1} P^{(1)}_S(S) = -1\;. $$ Here $P^{(1)}(S)$ is the randomized American put value and $\underline{S}_1 < K$ is the (unknown) optimal exercise boundary. Carr gives the following solution, which I am struggling to derive myself: \begin{align*} P^{(1)}(S) = \begin{cases} p^{(1)}(S) + b^{(1)}(S) \qquad & \text{if } S > K \\ KR - S + c^{(1)}(S) + b^{(1)}(S) \qquad & \text{if } S \in (\underline{S}_1, K) \\ K - S \qquad & \text{if } S \leq \underline{S}_1 \end{cases} \end{align*} Here, $p^{(1)}(S)$ is the randomized value of a European put paying $(K-S)^+$ at the first jump: $$ p^{(1)}(S) = \left(\frac{S}{K}\right)^{\gamma - \epsilon} (qKR - \hat{q}K), \qquad S > K $$ where $$ \gamma = \frac{1}{2} - \frac{r}{\sigma^2}, \qquad R = \frac{1}{1 + rT}, \qquad \epsilon = \sqrt{\gamma^2 + \frac{2}{R\sigma^2 T}} $$ and $$ p = \frac{\epsilon - \gamma}{2\epsilon}, \qquad q = 1-p, \qquad \hat{p} = \frac{\epsilon - \gamma + 1}{2\epsilon}, \qquad \hat{q} = 1 - \hat{p} \;. $$ The quantity $b^{(1)}(S)$ is described as the present value of interest received below the critical stock price $\underline{S}_1$ until the first jump, $$ b^{(1)}(S) = \left(\frac{S}{\underline{S}_1}\right)^{\gamma - \epsilon} qKRrT \;. $$ Finally, $c^{(1)}(S)$ is the randomized value of a European call paying $(S - K)^+$ at the first jump time, $$ c^{(1)}(S) = \left(\frac{S}{K}\right)^{\gamma + \epsilon} (\hat{p}K - pKR), \qquad S < K\;. $$

Here's what I did for the case $S \in (\underline{S}_1, K)$, for which I'm not arriving at Carr's solution. Using the change of variables $x \mapsto \log S$, it's easy to show the homogeneous solution to $(1)$ is $$ P^{(1)}_{homo}(S) = c_1 S^{\gamma + \epsilon} + c_2 S^{\gamma - \epsilon} $$ and that the particular solution is $$ P^{(1)}(S) = c_1 S^{\gamma + \epsilon} + c_2 S^{\gamma - \epsilon} + RK - S\;. $$ Using the boundary conditions $\lim_{S \, \downarrow \underline{S}_1} P^{(1)}(S) = K - \underline{S}_1$ and $\lim_{S \, \downarrow \underline{S}_1} P^{(1)}_S(S) = -1$, I get that \begin{align*} c_1 & = \underline{S}_1^{-\gamma - \epsilon}(pK - pKR), \\ c_2 & = \underline{S}_1^{\epsilon - \gamma}qKRrT \end{align*} This is almost the correct solution, if only we had $c_1 = K^{-\gamma - \epsilon}(\hat{p}K - pKR)$. That is, replacing $\underline{S}_1$ with $K$ and the first $p$ with $\hat{p}$. Can anyone spot where I may have made a mistake?

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