=+13. In the diffusion approximation to a branching process with immigration, we set (t, x)=( )x + and 2(t, x)=( + )x +

=+13. In the diffusion approximation to a branching process with immigration, we set μ(t, x)=(α − δ)x + ν and σ2(t, x)=(α + δ)x + ν, where α and δ are the birth and death rates per particle and ν is the immigration rate. Demonstrate that E(Xt) = x0eβt +

ν

β



eβt − 1



Var(Xt) = γx0(e2βt − eβt)

β + γν(e2βt − eβt)

β2

− γν(e2βt − 1)

2β2 +

ν(e2βt − 1)

2β

294 11. Diffusion Processes for β = α − δ, γ = α + δ, and X0 = x0. When α<δ, the process eventually reaches equilibrium. Find the limits of E(Xt) and Var(Xt).