16 Branching processes in continuous time L5 content chapter

Question 1

generating function for continous time branching process

Answer 1

We can also rearrange and take the limit ∆t → 0 to obtain a differential equation
for G_1 ≡ G_1(z, t) that turns out to be quite simple:
∂/∂t G_1
= −β(G_1 − φ(G_1))

≡ f(G1),

with G_1(z, 0) = z.
Clearly, the z−dependence only appears through the initial condition G1(z, 0) = z.