HJB equation then becomes $$\label{HJBone}
0 = \val2_t - \frac{1}{2}\frac{(\mu_i x \val2_x)^2}{\Sigma_i x^2 \val2_{xx}} - 1 - \log  \val2_x.$$ Make the ansatz $$\val2(x,t) = f_i(t)\epsilon_i\log x + g_i(t),$$ where $f_i$ and $g_i$ are to be determined and satisfy $f_i(T)=1$ and $g_i(T)=0$. Plug this into (\[HJBone\]), defining the constant $\widehat\rho_i = \mu_i^2\epsilon_i / 2\Sigma_i$, to get $$\left( \epsilon_i f_i'(t) + 1 \right) \log x + g_i'(t) + \widehat\rho_i f_i(t) - 1 - \log \epsilon_i - \log f_i(t) = 0.$$ Since there is only one term depending on $x$, we must have $$\epsilon_i f_i'(t) + 1 = 0, \quad\quad f_i(T)=1.$$ This yields $f_i(t) = \epsilon_i^{-1}(T - t) +1$. Then $g_i$ must solve $$g'_i(t)  =  - \widehat\rho_i (\epsilon_i^{-1}(T - t) +1) + 1 + \log ((T - t) + \epsilon_i),$$ which is easily integrated using $g_i(T)=0$ to get the solution $g_i$, though we will not need to use it explicitly. Note also that $f_i > 0$, and thus $w_x > 0$ and $w_{xx} < 0$ everywhere. We have therefore solved the HJB when $\delta_i = 1$. Recalling (\[optconsone\]), we deduce that the optimal controls are $$\begin{aligned}
\label{optcontrolOne}
\pi^{i,*} &= \frac{\mu_i}{\Sigma_i}, \quad\qquad c^{i,*}_t = \frac{1}{T - t + \epsilon_i}.\end{aligned}$$ Note that the result obtained 