Page 335 - Applied Probability

15. Diffusion Processes
324
It follows that
1
2

w(x)
= w(x)+ µ(x)w (x)s + σ (x)w (x)s

2

+E[g (T x ) | X 0 = x]s + o(s).
Rearranging this and sending s to 0 produce the differential equation
1 2

0= µ(x)w (x)+ σ (x)w (x)+ E[g (T x ) | X 0 = x].

2
The special cases g(t)= t and g(t)= e −θt correspond to the differential
equations
1 2

0= µ(x)w (x)+ σ (x)w (x) + 1 (15.11)

2
1 2

0= µ(x)w (x)+ σ (x)w (x) − θw(x), (15.12)
2
respectively.
Example 15.4.2 Fixation Times in the Neutral Model
In the diffusion approximation to the neutral Wright-Fisher model with
constant population size N, equation (15.11) becomes
x(1 − x)

0= w (x)+ 1. (15.13)
4N
If we take c = 0 and d = 1, then w(x) represents the expected time until
fixation of one of the two alleles. To solve equation (15.13), observe that
x
1
w (x) = −4N dy + k 1

1 y(1 − y)
2

1 1
x
= −4N + dy + k 1
1 y (1 − y)
2
= −4N [ln x − ln(1 − x)] + k 1
for some constant k 1 . Integrating again yields
x
w(x) = −4N [ln y − ln(1 − y)] dy + k 1 x + k 2
1
2
= −4N [x ln x +(1 − x)ln(1 − x)] + k 1 x + k 2
for some constant k 2 . The boundary condition w(0) = 0 implies k 2 =0,
and the boundary condition w(1) = 0 implies k 1 = 0. It follows that
w(x) = −4N [x ln x +(1 − x) ln(1 − x)] .
This is proportional to N and attains a maximum of 4N ln 2 at x =1/2.