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15. Diffusion Processes
323
and setting
v(x) − v(c)
u(x)=
(15.10)
.
v(d) − v(c)
Here the lower limit of integration l can be any point in the interval [c, d].
This particular solution also satisfies the boundary conditions.
Example 15.4.1 Fixation Probabilities in the Neutral Model
In the diffusion approximation to the neutral Wright-Fisher model with
constant population size N, we calculate
x
" y
v(x)= e − l 0 dz dy = x − l.
l
Thus, starting at a frequency of x for allele A 1 , allele A 2 goes extinct before
allele A 1 with probability
x − l − (c − l)
u(x) = lim = x.
c→0,d→1 d − l − (c − l)
This example is typical in the sense that u(x)= (x − c)/(d − c) for any
diffusion process with µ(x)= 0.
Another important problem is to calculate the expectation
w(x) = E[g(T) | X 0 = x]
n
of a function of the exit time T from [c, d]. For instance, g(t)= t gives the
nth moment of T, and g(t)= e −θt gives the Laplace transform of T.We
again derive an ordinary differential equation determining w(x), but now
the pertinent boundary conditions are w(c)= w(d)= g(0). To emphasize
the dependence of T on the initial position x, let us write T x in place of T.
We commence our derivation with the expansion
w(x) = E[g(T x ) | X 0 = x]
+ s) | X 0 = x]+ o(s)
=E[g(T X s
=E[g(T X s )+ g (T X s )s | X 0 = x]+ o(s)
=E{E[g(T X s ) | X s ] | X 0 = x} +E[g (T X s ) | X 0 = x]s + o(s)
=E[w(X s ) | X 0 = x]+ E[g (T x ) | X 0 = x]s + o(s).
Employing the same reasoning used in deriving the differential equation
(15.9) for u(x), we deduce that
1
2
E[w(X s ) | X 0 = x] = w(x)+ µ(x)w (x)s + σ (x)w (x)s + o(s).
2