Intuition

Given the jump times, $\{J_{n}, \ n\ge0\}$, and jump chain, $\{Y_{n}:n\ge0\}$, we usually want to recover $X_{t}$. Pick $t\ge0$, if $\exists n\ge0$ such that $J_{n}\le t<J_{n+1}$ then we have that $$X_{t}=Y_{n}$$But, this only works if $X_t$ is non-explosive or $P\left(\lim_{n\to\infty}J_{n}=\infty\right)=1$. So we’ll want to primarily deal with non-explosive processes going forward.

Intuition 1.5

For non-explosive We’re saying that as we count up the number of jumps (i.e. $n\to\infty$) then the jump times will also go up ($J_{n}\to\infty$). For explosive we’re saying that there’s some point where even though we increase the number of jumps to $\infty$ the time these occur are finite implying that our jump chain has exploded (which shown to be $\zeta$ below).

Intuition 2

What does “non-explosive” mean? Well essentially by inspecting the following figure: We see here $J_{i}$ explodes or diverges at $\zeta$ this shows that we make infinitely many jumps inside a finite interval. A non-explosive process does the opposite, it only jumps finitely many times inside any finite interval.