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Assume $(X_t)_{t\ge 0}$ is a homogeneous continuous-time Markov chain on the probability space $(\Omega, \mathcal{G}, \mathbb{P})$. Moreover, $X_1(\omega_0) = X_2(\omega_0)$ for some $\omega_0 \in \Omega$. Consider two random variables:

  • $I = \inf \left\{t \geq 1 \mid X_t \neq X_1\right\}$

  • $J = \inf \left\{t \geq 2 \mid X_t \neq X_2\right\}$

I would like to ask if we can conclude $I(\omega_0) = J(\omega_0)$. Thank you so much for your help!

Akira
  • 17,367
  • As random variables? Definitely not. Nor are they necessarily the same for this particular $\omega_0$. Nor do they even necessarily have the same distribution, unless you also assume that $X_3$ is the same (because effectively you have a kind of "Brownian bridge" between time $1$ and $2$ that you don't have after time $2$). – Ian Apr 29 '20 at 21:07
  • @Ian If I modify $I,J$ into $I = \inf \left{t \geq 1 \mid X_t \neq i\right}$ and $J = \inf \left{t \geq 2 \mid X_t \neq i\right}$. Can we obtain any improvement? – Akira Apr 29 '20 at 21:14
  • These are just the same thing, as long as $X_1(\omega_0)=i$. – Ian Apr 29 '20 at 22:09
  • Thank you so much for your answer @Ian! I will post your comment as answer to close this question. – Akira Apr 30 '20 at 06:39
  • @Ian Recently, I have asked some questions about continuous-time Markov chain, but none of them receives any attention. Because I'm doing my thesis, I hope that you can have a look at them. They are 1, 2, and 3. – Akira Apr 30 '20 at 06:39

1 Answers1

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I convert @lan's comment as answer to close this question.

As random variables? Definitely not. Nor are they necessarily the same for this particular $\omega_0$. Nor do they even necessarily have the same distribution, unless you also assume that $X_3$ is the same (because effectively you have a kind of "Brownian bridge" between time $1$ and $2$ that you don't have after time $2$).

Akira
  • 17,367