Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 27J / ii

Let $Z_t={\int }_0^{t}X(s)ds$ and set $a=q+\lambda$. Starting from $-1$, the first holding time has exponential distribution of rate $a$. Its joint density with a jump to $+1$ is $q{e}^{-au}du$. Before such a jump $Z$ decreases to $-u$, so the remaining rise needed to exceed $c$ is $c+u$. If instead the jump is to the absorbing state zero, the level stays nonpositive and never exceeds $c>0$.

Conditioning on this first jump and using the Strong Markov property gives