Consider a modification of the time-homogeneous version of the model by Black and Cox (Journal of Finance, 1976).

The firm value $V$ follows a Geometric Brownian motion: $$dV/V = \mu dt + \sigma dZ$$ with absorbing default boundary $V_b$ and constant coupon C.

With a constant interest rate $r$, the fundamental differential equation is an ordinary differential equation with bond price $F(V)$ satisfying: $$\frac{1}{2} \sigma^2 V^2 F_{vv}(V) + \mu V F_v(V) - rF(V) + C =0$$

Consider now the following modification. The interest rate takes the following form \begin{equation} r(V) = \begin{cases} r_1 \; \text{if } \; V \geq V_e \\ r_2 \; \text{otherwise} \end{cases} \end{equation} with $V_e > V_b$. Is it possible to derive a close form solution for the bond?

A few additional and preliminary thoughts. I would proceed in the following way: price a "bond" with two absorbing barriers $V_b$ and $V_e$ and constant interest rate $r_2$ using Kunitomo Ikeda (Mathematical Finance, 1992) and another bond using one default boundary $V_e$ and constant interest rate $r_1$, then sum the two pieces. Does the intuition make sense?

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