Poisson time-change representation of a Markov chain
= Poisson time-change representation of a Markov chain
{c}
{title2=$X(t)=X(0)+\sum_k\nu_kP_k(\int_0^t a_k(X(s-))\,ds)$}
= Random time-change representation
{synonym}
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.