Definition (Jump chain)

Given a continuous time process $\{X_{t}:t\in[0,\infty)\}$. A jump chain (or jump process), $Y_{n}, \ n\ge0$, is defined using jump times of $X_{t}$, $\{J_{n}:n\ge0\}$ i.e. $$Y_{n}=X_{J_{n}} \ \mbox{for }n\ge0$$