Page 348 - Applied Probability

15. Diffusion Processes
5. In the diffusion approximation to a branching process with immigra-
2
tion, we set µ(t, x)= (α−ν)x+η and σ (t, x)= (α+ν)x+η, where α
and ν are the birth and death rates per particle and η is the immigra-
tion rate. Justify these expressions by appealing to a continuous-time
Markov chain.
6. Continuing Problem 5, demonstrate that 337
E(X t )= x 0 e βt + η e βt − 1
β
βt
βt
γx 0 (e 2βt − e ) γη(e 2βt − e )
Var(X t )= +
β β 2
γη(e 2βt − 1) η(e 2βt − 1)
− +
2β 2 2β
for β = α − ν, γ = α + ν, and X 0 = x 0 . When α< ν, the process
eventually reaches equilibrium. Find the limits of E(X t ) and Var(X t ).
7. In Problem 5 suppose η = 0. Verify that the process goes extinct with
probability min{1,e −2 α−ν x 0 } by using equation (15.10) and sending
α+ν
c to 0 and d to ∞.
8. In Problem 5 suppose η> 0 and α< ν. Show that Wright’s formula
leads to the equilibrium distribution
4ην −1 2(α−ν)x
f(x) = k [(α + ν)x + η] (α+ν) 2 e α+ν
for some normalizing constant k> 0 and x> 0.

9. Consider the Wright-Fisher model with no selection but with muta-
tion from allele A 1 to allele A 2 at rate η 1 and from A 2 to A 1 at rate
η 2 . With constant population size N, prove that the frequency of the
A 1 allele follows the beta distribution
Γ[4N(η 1 + η 2 )] 4Nη 2 −1 4Nη 1 −1
f(x) = x (1 − x)
Γ(4Nη 2)Γ(4Nη 1 )
at equilibrium. (Hint: Substitute p(x)= x in formula (15.7) defining
2
the infinitesimal variance σ (t, x).)
10. Consider the transformed Brownian motion with infinitesimal mean
2
α and infinitesimal variance σ described in Example 15.2.2. If the
process starts at x ∈ [c, d], then prove that it reaches d before c with
probability
e −βx − e −βc 2α
u(x)= for β = .
e −βd − e −βc σ 2
Verify that u(x) reduces to (x − c)/(d − c) when α = 0. This simpli-
fication holds for any diffusion process with µ(x)= 0.