Cauchy-Euler ODE with indicator function in coefficient

Question

Consider the following Cauchy-Euler ODE, which is in particular the asset pricing equation for a (perpetual coupon defaultable) bond:

$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + C = 0$$

where $k \in \mathbb{R}_+$ and $$r = \begin{cases} r_1 \; \text{ if } \; V > k \\ r_2 \; \text{otherwise} \end{cases}$$

If $r$ was constant, the general solution has the form: $$F(V) = A_0 + A_1 V + A_2 V^{-x}$$ where $x \equiv \frac{m + \sqrt{m^2 + 2 r \sigma^2}}{\sigma^2}; \; \; m \equiv \mu - \frac{\sigma^2}{2}$ and the coefficients have to be determined using boundary conditions.

Is it possible to derive a similar general solution for the case of $r$ above? I'm hoping that one could solve separately the ODE over the two ranges separately and then paste the two solutions with a condition on the first derivative. Is this approach correct?

Answer 1

I solved it for the case $\mu = r_1$, the solution in $\mathbb{C}^1$ takes the guessed form $$F(V) = \begin{cases} A_0 + A_1 V + A_2 V^{-x} \; \text{ if } \; V>k \\ B_0 + B_1 V + B_2 V^{-y} \; \text{ else } \end{cases}$$ where the constants can be found by plugging the guess into the ODE and the remaining ones by imposing boundary conditions and smooth pasting at the discontinuity.