Page 328 - Applied Probability

15
Diffusion Processes
15.1 Introduction

The early application of diffusion processes by Fisher, Wright, and Kimura
elevated population genetics to one of the more sophisticated branches of
applied mathematics [3, 4, 8]. Although diffusion models address and solve
many interesting stochastic questions that are impossible to even discuss
in a deterministic framework, these models also raise the mathematical
bar. The current chapter surveys the theory at an elementary level, stress-
ing intuition rather than rigor. Readers with the time and mathematical
background should follow up this brief account by delving into serious pre-
sentations of the mathematics [1, 2, 7, 8].
Mathematical geneticists have pushed exact methods in diffusion models
about as far as one could realistically hope. The emphasis has been on sta-
tionary processes. Unfortunately, human genetics models need to take into
account population growth. If further progress is to be made in investigating
nonstationary models, then mathematical geneticists will have to pay more
heed to numerical methods. The final three sections of this chapter con-
front problems arising in numerical implementation of the Wright-Fisher
Markov chain and its diffusion approximation.

15.2 Review of Diffusion Processes

A diffusion process X t is a continuous-time Markov process that behaves
locally like Brownian motion. Its sample paths are continuous functions
confined to an interval I with left endpoint a and right endpoint b. In some
applications a = − or b =+∞ is appropriate. If a is finite, then I may be
either closed or open at a, and likewise at b. The process X t is determined
by the Markovian assumption and the distribution of its increments. For
small s and X t = x, the increment X t+s − X t is approximately normally
distributed with mean and variance

E(X t+s − X t | X t = x) = µ(t, x)s + o(s) (15.1)
2
2
E[(X t+s − X t ) | X t = x] = σ (t, x)s + o(s). (15.2)
2
The functions µ(t, x) and σ (t, x) ≥ 0 are called the infinitesimal mean
and variance, respectively. Here the term “infinitesimal variance” is used