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The stochastic process $\{X(t), t \geq 0\}$ is called homogeneous and will have stationary jump probability if $$P_{ij}(s, t)=P(x(t)=j \mid x(s)=i))=P_{ij}(0, t-s), \forall t, s \geq 0$$

We write $P_{ij}(t-s)$ for the probabilities $P_{ij}(s, t)$.

The Chapman-Kolmogorov equations are written as $$P_{ij}(t+s)=\sum_{k=0}^\infty P_{kj}(t)P_{kj}(s), \forall t, s \geq 0$$

Can you explain to me why at the Chapman-Kolmogorov equations we have a formula for $P_{ij}(t+s)$ and before we had $P_{ij}(t-s)$ ???

Why does it stand that at the Chapman-Kolmogorov equations at $P_{kj}(t)$ the index is $kj$ and not $ik$ ???

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    Seems like a typo. Also, what is $m$? – Math1000 Feb 16 '15 at 23:57
  • So you mean that $P_{ij}(t+s)$ is a typo and it should be $P_{ij}(t-s)$ ??? $m$ shouldn't exist... It is a typo... @Math1000 –  Feb 17 '15 at 00:03
  • Oh, I meant that it should be $ik$ and not $kj$ in the first term in the sum. – Math1000 Feb 17 '15 at 00:26
  • Ok. Thank you! Do you maybe have also an idea for: http://math.stackexchange.com/questions/1151298/the-chain-jumps-from-the-situation-i-to-j @Math1000 ??? –  Feb 17 '15 at 00:29

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