I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for a defaultable bond: $$ dP(t) = r(t)P(t)dt - P(t)dJ(t), $$ where $r(t)$ is an adapted process, and $J(t)$ is a jump process that changes from zero to one on default of the bond issuer.
I'm trying to solve this SDE by finding a closed form formula for $P(t)$, where I'm following the theory given in Steven Shreve's book: -Stochastic Calculus for Finance, Continuous-Time Models- (Chapter 11). I'm attempting to use Ito's formula for jumps, but I'm stuck. Any hints on how to proceed to formally get $P(t)$ from the SDE? Thanks in advance.