Page 339 - Applied Probability

15. Diffusion Processes
328
2N(1−f)
z
2
k 4
=
4N(1 − f)
∞
m
+2Nη−1 −z
e
dz
z 2
≈
2[2N(1 − f)] 2
+2Nη)
k 4 Γ( m k 4 0 m +2Nη
0 2N(1 − f) m+4Nη−2 e −z dz
2
= m +2Nη .
2[2N(1 − f)] 2
Taking m = 0 identifies the normalizing constant
2[2N(1 − f)] 2Nη
k 4 = .
Γ(2Nη)
With this value of k 4 in hand, the mean of f(x)is
1
Γ(2Nη + )
2
xf(x) dx ≈ .
I 2N(1 − f)Γ(2Nη)
When Nη is large, application of Stirling’s formula implies that the mean

is close to the deterministic equilibrium value η/(1 − f). In practice, one
should be wary of applying the equilibrium theory because the approach
to equilibrium is extremely slow.
15.7 Numerical Methods for Diffusion Processes
It is straightforward to simulate a diffusion process X t . The definition tells
us to extend X t to X t+s by setting the increment X t+s −X t equal to a nor-
2
mal deviate with mean µ(t, x)s and variance σ (t, x)s. The time increment
s should be small, and each sampled normal variate should be independent.
Techniques for generating random normal deviates are covered in standard
texts on computational statistics and will not be discussed here [9, 10].
Of more concern is how to cope with a diffusion process with finite range
I. Because a normally distributed random variable has infinite range, it
is possible in principle to generate an increment that takes the simulated
process outside I. One remedy for this problem is to take s extremely small.
2
It also helps if the infinitesimal variance σ (t, x) tends to 0 as x approaches
the boundary of I. This is the case with the neutral Wright-Fisher process.
Simulation offers a crude method of finding the distribution of X t .Sim-
ply conduct multiple independent simulations and compute a histogram
of the recorded values of X t . Although this method is neither particu-
larly accurate nor efficient, it has the virtue of yielding simultaneously the
distributions of all of the X t involved in the simulation process. Thus, if
1000 times are sampled per simulation, then the method yields all 1000
distributions, assuming that enough computer memory is available. Much