Page 333 - Applied Probability

15. Diffusion Processes
322
lation geneticists substitute p(x)= x in formula (15.7) defining the infin-
2
itesimal variance σ (t, x). This action is justified for neutral and recessive
inheritance, but less so for dominant inheritance where the allele frequency
x is typically on the order of magnitude of the mutation rate η.It isalso
fair to point out that in the presence of inbreeding or incomplete mixing of
a population, the effective population size is less than the actual pop-
ulation size [3]. For the sake of simplicity, we will ignore this evolutionary
fact.
15.4 First Passage Time Problems
Let c<d be two points in the interior of the range I of a diffusion process
X t . Define T c to be the first time t that X t = c and similarly for T d. The
process X t exits (c, d) at the time T =min{T c,T d}. We consider two related
problems involving these first passage times. One problem is to calculate
the probability u(x) = Pr(T d <T c | X 0 = x) that the process exits via d
starting from x ∈ [c, d]. It is straightforward to derive a differential equation
determining u(x) given the boundary conditions u(c) = 0 and u(d)= 1.
With this end in mind, we assume that X t is time homogeneous.
For s> 0 small and x ∈ (c, d), the probability that X t reaches either c
or d during the time interval [0,s]is o(s). Thus,
u(x)=E[u(X s ) | X 0 = x]+ o(s).

If we let ∆X s = X s −X 0 and expand u(X s ) in a second-order Taylor series,
then we find that

u(X s )= u(x +∆X s )
1 2
= u(x)+ u (x)∆X s + u (x)+ r(∆X s ) ∆X , (15.8)

s
2
where the relative error r(∆X s ) tends to 0 as ∆X s tends to 0. Invoking
equations (15.1), (15.2), and (15.8) therefore yields
u(x) = E[u(X s )] + o(s)
1 2

= u(x)+ µ(x)u (x)s + σ (x)u (x)s + o(s),
2
which upon rearrangement and sending s to 0 gives the differential equation
1 2
0= µ(x)u (x)+ σ (x)u (x). (15.9)

2
It is a simple matter to check that equation (15.9) can be solved explicitly
by defining
x
" y 2µ(z)
v(x) = e − l σ 2 (z) dz dy
l