= First-jump decomposition of a piecewise-deterministic Markov process
{title2=$P_{s,t}=P^\alpha_{s,t}+\int_s^tP^\alpha_{s,u}\alpha K_uP_{u,t}\,du$}
For a <piecewise-deterministic Markov process>, conditional on no jump before $u$, the state is $z(u)=\varphi(s,x,u)$ 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>.
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