 Solution

For a time-independent value, $\lambda =r$. This is an Euler equation. Put $f=S^{\alpha }$ to obtainFor $\sigma >0$, let $b_0=r-q-{\sigma }^2/2$. The roots areThey are distinct and real because $r>0$, with one positive and one negative. The general stationary solution on $S>0$ is $\boxed{V(S)=A{S}^{{\alpha }_{+}}+B{S}^{{\alpha }_{-}}}$. There are no boundary restrictions in this subpart to eliminate either coefficient. In the degenerate case $\sigma =0$, the equation is first order: for $r\ne q$ its solutions are $A{S}^{r/(r-q)}$, while for $r=q$ only the zero stationary solution remains.