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For a birth and death process with birth rates, $\lambda_i$ and death rates $\mu_i$ $(i=0,1,2...)$ respectively. Show that the transition probabilities, $P_{i,j}(t)$ satisfy the following differential equations

$P'_{0,j}(t) = -\lambda_{0} P_{0,j}(t) +\lambda_{0} P_{1,j}(t)$

$P'_{i,j}(t) = \mu_{i} P_{i-1,j}(t)-(\lambda_{i} +\mu_{i}) P_{i,j}(t) +\lambda_{i}P_{i+1,j}(t)$ for $i \geq 1$

Here is what I have so far:

First of all, the subscripts are different than a normal birth and death process. So I think the transition matrix is the following

$P_{i,j}(t) = $\begin{matrix} -\lambda_{0} & \lambda_{0} & 0 & 0 & ... \\ \mu_{0} & -(\lambda_{1}+\mu_{0}) & \lambda_{1} & 0 &...\\ 0 & \mu_{1} & -(\lambda_{2} + \mu_{1}) & \lambda_{2} &0 &...\\ 0 & 0 & \mu_{2} & -(\lambda_{3} + \mu_{2} ) & \lambda_{3} &...\\ \end{matrix}

I know from the Chapman-Kolomogrov Equations, I can somehow manipulate them in order to get the differential equations but I am not exactly sure how to do that.

Thank you for all of your inputs. Much help is appreciated.

  • Is that really all you are given? Without defining what "birth rate" and "death rate" are , it would be unnecessarily difficult to help. If this is markov system, then what are your states? Is this discrete time? Discrete space? – DanielV May 11 '15 at 03:36
  • Oh sorry. This is supposed to be for a continuous time and space markov chains with infinite capacity. –  May 11 '15 at 03:47
  • Can you give a mathematical definition of $\lambda$ and $\mu$ and $P$? – DanielV May 11 '15 at 03:57
  • I am not sure what you mean but I defined $\lambda$ as birth rates and $\mu$ as death rates. The matrix I inputed is P. Sorry it looks kind of weird. However the death rates are labeled differently. –  May 11 '15 at 04:02
  • @DanielV Your desire of completeness is laudable but really these are standard definitions of the field. – Did May 14 '15 at 10:42

1 Answers1

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We begin with the Chapman-Kolmogorov equations. For small $h$, we have

\begin{eqnarray*} P_{i,j}(t+h) &=& \sum_{k=0}^{\infty} P_{i,k}(h) P_{k,j}(t) \\ && \\ &=& P_{i,i-1}(h) P_{i-1,j}(t) + P_{i,i}(h) P_{i,j}(t) + P_{i,i+1}(h) P_{i+1,j}(t) + o(h) \\ && \qquad\text{since the probability of transition in time $h$ from $i$ to} \\ && \qquad\text{anyhere other than $i-1,\;i,\;i+1$ equals $o(h)$.} \\ && \\ &=& \mu_i h P_{i-1,j}(t) + (1-\lambda_i h-\mu_i h) P_{i,j}(t) + \lambda_i h P_{i+1,j}(t) + o(h) \\ && \\ \therefore\quad \dfrac{P_{i,j}(t+h) - P_{i,j}(t)}{h} &=& \mu_i P_{i-1,j}(t) - (\lambda_i + \mu_i) P_{i,j}(t) + \lambda_i P_{i+1,j}(t) + o(h) \\ && \\ \therefore\quad P_{i,j}^{'}(t) &=& \mu_i P_{i-1,j}(t) - (\lambda_i + \mu_i) P_{i,j}(t) + \lambda_i P_{i+1,j}(t) \\ && \qquad\text{taking the limit as $h\to0$.} \end{eqnarray*}

For the particular case of $i=0$, where $\mu_i=0$, this equation becomes

\begin{eqnarray*} P_{0,j}^{'}(t) &=& - \lambda_0 P_{0,j}(t) + \lambda_0 P_{1,j}(t). \end{eqnarray*}

Did
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