 Solution

A path that reaches level $c$ from below does so while in state $+1$, because $Z$ is continuous and has slope $X$. At the first crossing time, the residual exponential holding time is memoryless and the Strong Markov property restarts the chain from $+1$. Conditional on ever reaching $c$, the probability of subsequently exceeding $c+d$ is therefore exactly the original probability of exceeding $d$. HenceThe zero-level limit is one because the initial $+1$ holding time is positive almost surely. Also $0<{\psi }_{+}(c)\le 1$, and it is nonincreasing. Consequently $h(c)=-\log {\psi }_{+}(c)$ is nondecreasing and additive. First $h$ is linear on positive rationals, then monotonicity gives the same result on all positive reals. ThusAbsorption can be viewed as killing at rate $\lambda$, independent of sign changes at rate $q$; this ensures the maximum is finite almost surely and makes the positive decay rate found in (iv) natural.