Given the infinitesimal generator, how a continuous Markov chain behaves after the exploding time?

Let $$\mathbb X$$ be the state space and $$\Gamma=(\Gamma_{x y})_{x, y \in X}$$ the infinitesimal generator of the continuous Markov chain. The stochastic Markovian matrix $$P=(P_{x y})_{x, y \in \mathbb X}$$ is defined by $$P_{x y}=\left\{\begin{array}{ll} \frac{\Gamma_{x y}}{\gamma_{x}} & \text { if } \gamma_{x} \neq 0 \\ 0 & \text { if } \gamma_{x}=0 \end{array} \text { for } y \neq x, \text { and } P_{x x}=\left\{\begin{array}{ll} 0 & \text { if } \gamma_{x} \neq 0 \\ 1 & \text { if } \gamma_{x}=0 \end{array}\right.\right.$$

The kernel $$P$$ defines a discrete-time $$(\mathbb X, P)$$-Markov chain $$\tilde{\xi}=(\tilde{\xi}_{n})_{n \in \mathbb{N}}$$ termed the Markov chain embedded at the moments of jumps. Define a sequence $$\sigma=(\sigma_{n})_{n \geq 1}$$ of random holding times distributed, conditionally on $$\tilde{\xi}$$, according to an exponential law. More precisely, consider $$\mathbb{P}\left(\sigma_{n} \in \mathrm{d} s | \tilde{\xi}\right)=\gamma_{\tilde{\xi}_{n-1}} \exp \left(-s \gamma_{\tilde{\xi}_{n-1}}\right) \mathbf{1}_{\mathbb{R}_{+}}(s) \,\mathrm{d} s$$ so that $$\mathbb{E}\left(\sigma_{n} | \tilde{\xi}\right)=1 / \gamma_{\tilde{\xi}_{n-1}}$$.

The sequence $$J=(J_{n})_{n \in \mathbb{N}}$$ of random jump times is defined accordingly by $$J_{0}=0$$ and for $$n \geq 1$$ by $$J_{n}=\sum_{k=1}^{n} \sigma_{k}$$. The life time is denoted $$\zeta=\lim _{n \rightarrow \infty} J_{n}$$. To have a unified description of both explosive and non-explosive processes, we can extend the state space into $$\hat{\mathbb X}=\mathbb X \cup\{\partial\}$$ by adjoining a special absorbing state $$\partial$$. The continuous-time Markov chain is then the càdlàg process $$\xi=(\xi_{t})_{t \in[0, \infty]}$$ defined by $$\xi_{0}=\tilde{\xi}_{0} \text { and } \xi_{t}=\left\{\begin{array}{ll} \sum_{n \in \mathbb{N}} \tilde{\xi}_{n} \mathbf{1}_{[J_{n}, J_{n+1})}(t) & \text { for } 0

Let $$(X_t)_{t \in[0, \infty]}$$ be the Markov chain defined by $$(\mathbb X, \Gamma)$$.

In case $$\zeta < \infty$$, it seems to me that we don't know how the $$X_t$$ behaves when $$t \ge \zeta$$, so the authors introduce $$\partial$$. This goes against my understanding because we are given $$(\mathbb X, \Gamma)$$ and thus we know $$(X_t)_{t \in[0, \infty]}$$.

Could you pleas elaborate on my confusion?

• If $\zeta<\infty$ then $(\mathbb{X},\Gamma)$ and the given entropy $\omega$ cannot specify what $X_\zeta(\omega)$ is, so you do not know $(X_t)_{t \in [0,\infty)}$. Intuitively, the system changes state infinitely often in an arbitrarily small interval to the left of $t=\zeta$ so there is no way to settle on what $X_\zeta$ should actually be. The convention is to say that at time $\zeta$ the process "exploded" and use this auxiliary state (with trivial dynamics associated to it) to track paths that have already exploded at various times.
– Ian
May 4, 2020 at 16:04
• @Ian so the it maybe the case that the CTMC $(X_t)_{t \in[0, \infty]}$ enduces $(\mathbb{X},\Gamma)$, but this $(\mathbb{X},\Gamma)$ can not give information about the chain after $\zeta$ in case $\zeta<\infty$, right? As such, it maybe the case that $(\mathbb X, \Gamma)$ is not sufficient to determine its associated CTMC . May 4, 2020 at 16:10
• Yes, if a chain can explode then $(\mathbb{X},\Gamma)$ does not determine it for all time. Basically it determines it for any finite number of jumps, but if infinitely many jumps can occur in finite time then the generator ceases to specify the process.
– Ian
May 4, 2020 at 16:10
• @Ian Thank you so much for clearing out my biggest confusion in recent weeks! I get this confusion because many lecture notes on the Internet said that giving $(\mathbb{X},\Gamma)$ is equivalent to giving a CTMC. May 4, 2020 at 16:13
• I think your confusion rests in the question of what it means to specify a possibly-exploding CTMC, which is to specify the dynamics until explosion. After explosion, anything you do thereafter is more or less arbitrary.
– Ian
May 4, 2020 at 17:07

If $$\zeta<\infty$$ then $$(\mathbb{X},\Gamma)$$ and the given entropy $$\omega$$ cannot specify what $$X_\zeta(\omega)$$ is, so you do not know $$(X_t)_{t \in [0,\infty)}$$. Intuitively, the system changes state infinitely often in an arbitrarily small interval to the left of $$t=\zeta$$ so there is no way to settle on what $$X_\zeta$$ should actually be. The convention is to say that at time $$\zeta$$ the process "exploded" and use this auxiliary state (with trivial dynamics associated to it) to track paths that have already exploded at various times.
Yes, if a chain can explode then $$(\mathbb{X},\Gamma)$$ does not determine it for all time. Basically it determines it for any finite number of jumps, but if infinitely many jumps can occur in finite time then the generator ceases to specify the process.