Additive functionals and a time change which preserves the semi-Markov property of a process

Stochastic processes with paths belonging to $D(\ell_+\to X)$ ($X$ is a metric space) and their time change transformations are considered. It is proved that the necessary and sufficient condition for this transformation to be preserving the semi-Markov property of the processes is the possibility to construct a time change with a family of additive functionals ($a_\tau(\lambda)$, $\lambda\ge0$, $\tau\in\mathscr T$), где
$$ \exp(-a_\tau(\lambda))=\int_0^\infty\exp(-\lambda t)F_\tau(dt), $$
$F_\tau$ – being the condition distribution of stopping time $\tau$ of the transformed process and $\mathscr T$ is a family of the first exit times from open sets and their iterations.