Probability transitions for branching processby kai_sikorski Tags: branching, probability, process, transitions 

#1
Feb2712, 01:30 PM

PF Gold
P: 162

Working through a paper that uses this result about branching processes. Can't seem to figure out a way to connect the dots. Anyone have any suggestions?
Let Z_{t} be a branching process where each individual gives birth to a single offspring at rate β > 1 and dies at rate 1. The transition probabilities for Z_{t} can be computed exactly and are given by [tex]P_{10}(t) = \frac{1  e^{(\beta1)t}}{\beta  e^{(\beta1)t}}[/tex] [tex]P_{1k}(t) = (1  P_{10}(t))(1  \eta(t))\eta(t)^{k1}[/tex] [tex]\eta(t) = \frac{1  e^{(\beta1)t}}{1  \frac{1}{\beta}e^{(\beta1)t}}[/tex] I know that P_{1i} satisfy the following system of equations [tex] P_i'(t) = Q P(t) [/tex] Where Q is the transition rate matrix (i think that's what it's called) and is given by [tex] Q = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & \\ 0 & (1+\beta ) & 2 & \ddots & \vdots \\ 0 & \beta & 2(1+\beta ) & 3 & \\ \vdots & \ddots & 2\beta & 3(1+\beta)& \ddots \\ & \ldots & & \ddots & \ddots \\ \end{array} \right) [/tex] I have verified the above solution works, but I'm not sure how one would have gotten it. Any ideas? 



#3
Feb2812, 03:07 AM

PF Gold
P: 162

Oh I figured it out, mostly
Let G be the generating function for Z_{t} [itex] G[s,t] = p_{10}(t) + p_{11}(t)s + p_{12}(t)s^2 + ... [/itex] By using the differential equations for the ps, and manipulating some indices you can get a PDE for G [itex] \partial_t G[s,t](1s(1+\beta))+\beta s^2) \partial_s G[s,t] = 0[/itex] [itex] G[s,0] = s [/itex] I can plug this into DSolve and it gives the right answer. However I'm having some trouble doing the method of characteristics on this. For the family of characteristics I get [itex] s[t] = \frac{e^{t \beta +c}+e^{t+\beta c}}{e^{t \beta +c}+e^{t+\beta c} \beta }[/itex] But I don't know how to pick c, so that I can set an initial condition on s[t]. Any ideas? 


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