A continuous markov process $(X_{t})_{t\geq_{0}}$ with corresponding Q matrix Q defined on a countable space $I$ is said to be explosive if $\mathbb{P}_{i}(\xi < \infty) >0$ for some $i \in I$ where $\xi = \underset{ n }{ \text{sup} }J_{n}$, where $J_{n}$ is the jumping time.
My question is if we fix the initial state, does there exist a continuous Markov chain with a corresponding Q matrix that explodes with probability p, s.t $0 < p < 1$
Note here that we have conditioned on the initial state of the Markov chain, if we furthermore fix the trajectory, the Markov chain either explodes with probability 0 or 1 by some properties of exponential random variables.
I asked this question to my professor, and he responded by saying that we can consider a 3-dimensional random walk, which starts at some states that are not the origin. He said that we can kill the process once it reaches zero, and since the probability for the process to reach zero is between 0 and 1, we are done. However, I'm really confused about what he meant by that killing this process would result in an explosion, isn't $\xi$, the explosion time defined as $\underset{ n }{ \text{sup } } J_{n}$?
Best Answer
Recall that in a pure-birth process, explosion occurs if and only if $\sum_{n=0}^\infty\lambda_n^{-1}<\infty$. Consider a CTMC on the integers with $\lambda_n := Q_{n,n+1}$ for $n\geqslant 0$ and $\mu_n:= Q_{n,n-1}$ for $n\leqslant 0$ with $\sum_{n=0}^\infty \lambda_n^{-1}<\infty$ but $\sum_{n=0}^\infty \mu_n^{-1}=+\infty$.
Conditioned on the initial state being $0$, then explosion occurs with probability $\frac{\lambda_0}{\lambda_0+\mu_0}$, which lies in $(0,1)$ assuming $\lambda_0,\mu_0>0$.