Question

Find the generating function \(\varphi(t ; s)\) of the continuous time branching process if the infinitesimal generating function is $$ u(s)=1-s-\sqrt{1-s} $$

Short answer

The generating function \(\varphi(t ; s)\) of the continuous time branching process with infinitesimal generating function \(u(s) = 1-s-\sqrt{1-s}\) is given by: $$ \varphi(t ; s) = \int \frac{1}{(1-s-\sqrt{1-s})} \left[\frac{2}{3}(1-s)^{\frac{3}{2}} - (1-s)^2\right]ds $$

Step-by-step solution

Rewrite the Kolmogorov's Forward Equation using the given u(s)

We are given the infinitesimal generating function \(u(s)=1-s-\sqrt{1-s}\). Plug this into the forward equation: $$ \frac{\partial \varphi(t ; s)}{\partial t} = u\left(\frac{\partial \varphi(t ; s)}{\partial s}\right) = \left(1-s-\sqrt{1-s}\right)\frac{\partial \varphi(t ; s)}{\partial s} $$

Solve the equation for \(\varphi(t;s)\) using the Method of Characteristics

To solve the above equation using the method of characteristics, let's introduce the new variables: $$ \frac{dz}{dt} = 1-s-\sqrt{1-s} \qquad \text{and} \qquad \frac{ds}{dt} = \frac{\partial \varphi(t ; s)}{\partial s} $$ Now, solve the first equation for \(z\): $$ \frac{dz}{ds} = \frac{dt}{ds} \cdot \frac{dz}{dt} = \frac{1}{\frac{\partial \varphi}{\partial s}}(1-s-\sqrt{1-s}), $$ where we have used the relation \(\frac{ds}{dt} = \frac{\partial \varphi}{\partial s}\). We can rewrite it as $$ \frac{\partial \varphi}{\partial s} = \frac{1}{(1-s-\sqrt{1-s})} \frac{dz}{ds} $$ Now we can integrate both sides with respect to \(s\): $$ \varphi(t ; s) = \int \frac{1}{1-s-\sqrt{1-s}} dz(s) $$

Obtain the generating function \(\varphi(t; s)\)

To obtain \(\varphi(t;s)\), we need to solve the integral for \(z(s)\): $$ z(s) = \int (1-s-\sqrt{1-s}) ds $$ Let \(w = 1-s, dw = -ds\), then the integral becomes $$ z(w) = \int (w-\sqrt{w}) dw $$ By integrating, we get $$ z(w) = \frac{2}{3}w^{\frac{3}{2}} - w^2 = \frac{2}{3}(1-s)^{\frac{3}{2}} - (1-s)^2 $$ Now plug \(z(s)\) back into the expression for \(\varphi(t;s)\): $$ \varphi(t ; s) = \int \frac{1}{(1-s-\sqrt{1-s})} dz(s) = \int \frac{1}{(1-s-\sqrt{1-s})} \left[\frac{2}{3}(1-s)^{\frac{3}{2}} - (1-s)^2\right]ds $$ This is the generating function \(\varphi(t; s)\) of the continuous time branching process.