First-jump decomposition of a piecewise-deterministic Markov process ($P_{s,t}=P_{s,t}^{\alpha }+{\int }_s^t{P}_{s,u}^{\alpha }\alpha K_u{P}_{u,t}du$)

For a piecewise-deterministic Markov process, conditional on no jump before $u$, the state is $z(u)=\phi (s,x,u)$ and the survival probability is $S(u)=\exp [-{\int }_s^u\alpha (z(r))dr]$. The first jump has probability density function $\alpha (z(u))S(u)$; conditioning on its time and its destination gives $P_{s,t}f(x)=S(t)f(z(t))+{\int }_s^t\alpha (z(u))S(u)\int P_{u,t}f(y)K_u(z(u),dy)du$. This proves the identity for bounded Borel measurable functions under nonexplosion.