Poisson time-change representation of a Markov chain ($X(t)=X(0)+{\sum }_k{\nu }_k{P}_k({\int }_0^t{a}_k(X(s-))ds)$)

A jump process with event types $k$, increments ${\nu }_k$, and rates $a_k(X)$ can be represented as $X(t)=X(0)+{\sum }_k{\nu }_k{P}_k({\int }_0^t{a}_k(X(s-))ds)$, with independent unit-rate Poisson processes. This represents the Markov jump-process generator through time-changed counts.