Non-Explosive

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Definition

StochasticProcesses

A continuous-time process $\{X_{t}:t\ge0\}$ is non-explosive if $P\left(\lim_{n\to\infty}J_{n}=\infty\right)=1$ and explosive otherwise. # 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: Pasted image 20231128125934.png 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.

Linked from

Continuous Time Markov Chain

Poisson Process