Page 343 - Applied Probability

15. Diffusion Processes
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c ij→i+1,k p ij→i+1,k based on elementary functions and the standard normal
distribution function. The pieces p ij→i+1,k and c ij→i+1,k are then reassem-
bled into probabilities and centers of probabilities using equations (15.20).
at time t i is governed by
Choice of the mesh points a i0 < ··· <a i,r i
several considerations. First, the probability Pr(X t i ∈ [a i0 ,a i,r i ]) should be
2
negligible. Second, σ (t i ,x) should be well approximated by a constant and
µ(t i ,x) by a linear function on each interval [a ij ,a i,j+1 ]. Third, the density
f(t i ,x) should be well approximated by a linear function on [a ij ,a i,j+1 ]as
well. This last requirement is the hardest to satisfy in advance, but nothing
prevents one from choosing the next subdivision adaptively based on the
distribution of probability within the current subdivision. Adding more
mesh points will improve accuracy at the expense of efficiency. Mesh points
need not be uniformly spaced. It makes sense to cluster them in regions
of high probability and rapid fluctuations of f(t, x). Given the smoothness
expected of f(t, x), rapid fluctuations are unlikely.
Many of the probabilities p ij→i+1,k are negligible. We can accelerate the
algorithm by computing p ij→i+1,k and c ij→i+1,k only for [a i+1,k ,a i+1,k+1 ]
is nor-
close to [a ij ,a i,j+1 ]. Because the conditional increment X t i+1 − X t i
mally distributed, it is very unlikely to extend beyond a few standard de-
is in [a ij ,a i,j+1 ]. Thus, the most sensible strategy is
viations σ ij given X t i
to visit each interval [a ij ,a i,j+1 ] in turn and propagate probability only to
those intervals [a i+1,k ,a i+1,k+1 ] that lie a few standard deviations to the
left or right of [a ij ,a i,j+1 ].

15.8 Numerical Methods for the Wright-Fisher
Process

One of the problems with the diffusion approximation to the Wright-Fisher
Markov chain is that it degrades for very low allele frequencies. Because of
the interest in gene extinction, this is regrettable. However in the regime of
low allele frequencies, we can always fall back on the Wright-Fisher Markov
chain. As population size grows, the Markov chain updates become more
and more computationally demanding. The interesting issue thus becomes
how to merge the Markov chain and diffusion approaches seamlessly into a
single algorithm for following the evolution of an allele. Here we present one
possible algorithm and apply it to understanding disease-gene dynamics in
a population isolate.
The algorithm outlined in the previous section has the virtue of being
posed in terms of distribution functions rather than density functions. For
low allele frequencies, discreteness is inevitable, and density functions are
unrealistic. In adapting the algorithm to the regime of low allele frequencies,