Page 347 - Applied Probability

15. Diffusion Processes
336
and the standard deviation increases only very slightly. Less severe bot-
tlenecks at generation 57 are unnoticeable in these plots. Of course, if a
bottleneck of this fractional magnitude were introduced earlier, then the
impact would be substantial.
15.10 Problems
1. Consider a diffusion process X t with infinitesimal mean µ(t, x) and
2
infinitesimal variance σ (t, x). If the function f(t) is strictly increas-
ing and continuously differentiable, then argue that Y t = X f(t) is a
diffusion process with infinitesimal mean and variance
µ Y (t, y)= µ[f(t),y]f (t)

2
2
σ (t, y)= σ [f(t),y]f (t).

Y
Apply this result to the situation where Y t starts at y 0 and has
2
2
µ Y (t, y)=0 and σ (t, y)= σ (t). Show that Y t is normally dis-
Y
tributed with mean and variance
E(Y t ) = y 0
t

2
Var(Y t ) = σ (s) ds.
0
(Hint: Let X t be standard Brownian motion.)
2. Consider a time-homogeneous diffusion process X t starting at x 0 and
2
2
having µ(t, x)= −αx+η and σ (t, x)= σ . Show that X t is normally
distributed with mean and variance
η(1 − e −αt )
−αt
E(X t )= x 0 e +
α
2
σ (1 − e −2αt )
Var(X t )= .
2α
The case η = 0 and α> 0 is the Ornstein-Uhlenbeck process. (Hints:
The transformed process Y t = X t/σ − η/α has infinitesimal mean
2
2
2
µ Y (t, z)= −αx/σ and infinitesimal variance σ (t, z) = 1. Check
Y
Kolmogorov’s forward equation for Y t .)
3. Prove that the diffusion process X t discussed in Problem 2 is not a
smooth, invertible transformation X t = g(t, Y t ) of standard Brownian
motion Y t .
4. Calculate the equilibrium distribution for the diffusion process dis-
cussed in Problem 2 by applying Wright’s formula (15.17). What
restriction must you place on α? Show that your conclusions are con-
sistent with the limiting mean and variance of the process.