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A Markov chain in discrete time is irreducible, has state space $\{0,1,\dots\}$ and starts at $1$. It is both a branching process and a martingale. Determine the probability of hitting $0$.

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    $\begingroup$ what did you try in solving this problem? $\endgroup$
    – Ilya
    May 23, 2013 at 13:37
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    $\begingroup$ Where did you get the problem? Why does it interest you? What do you know about Markov chains, about irreducibility, about branching processes, about martingales? $\endgroup$ May 23, 2013 at 13:42
  • $\begingroup$ Nice question (the answer is $1$). It is a pity that this user is mishandling the site to such an extent... $\endgroup$
    – Did
    Jul 4, 2013 at 17:28

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Under the assumption that this is a birth-and-death reversible MC (rate of inflow=rate of outflow for each state), and the transition probabilities are $p,q,r$ (to go up one state/down one state, remain in the same state) with $p+q+r=1$ and boundary value $h_{0,0}=1$ we can get the following equation: $$ h_{1,0}=qh_{0,0,}+rh_{1,0}+ph_{2,0}\\ p h_{1,0}=qh_{2,0} $$ The second equation come from the reversibility property (detailed balance equation). Solving these, we get: $$ h_{1,0}=\frac{q^2}{p^2-q^2+pq} $$

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  • $\begingroup$ Your question was to determine the probability of 'hitting 0' without providing any specific details. This is this probability under some specific assumptions on the MC $\endgroup$
    – Alex
    May 23, 2013 at 16:25
  • $\begingroup$ Offtopic. The hypotheses made in this answer contradict the assumption that the process is both a branching process and a martingale. $\endgroup$
    – Did
    Jul 4, 2013 at 17:27

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