Page 329 - Applied Probability

15. Diffusion Processes
318
rather than “infinitesimal second moment” because the approximation
Var(X t+s − X t | X t = x)
2
=E[(X t+s − X t ) | X t = x] − [µ(t, x)s + o(s)]
2
=E[(X t+s − X t ) | X t = x]+ o(s) 2
follows directly from approximations (15.1) and (15.2). If the infinitesimal
mean and variance do not depend on time t, then the process is time
2
homogeneous.If µ(t, x) = 0 and σ (t, x) = 1, then X t reduces to standard
Brownian motion.
To begin our nonrigorous, intuitive discussion of diffusion processes, we
note that the normality assumption implies
4 4
√ m
m 4
m 4 X t+s − X t 4 4
E(|X t+s − X t | | X t = x)= E 4 √ 4 4 X t = x σ(t, x) s
4 σ(t, x) s 4
= o(s) (15.3)
for m> 2. This insight is crucial in various arguments involving Taylor
series expansions. For instance, it allows us to deduce how X t behaves
under a smooth, invertible transformation. If Y t = g(t, X t ) denotes the
transformed process, then
∂ ∂ 1 ∂ 2 2
Y t+s − y = g(t, x)s + g(t, x)(X t+s − x)+ g(t, x)s
∂t ∂x 2 ∂t 2
∂ 2 1 ∂ 2 2
+ g(t, x)s(X t+s − x)+ g(t, x)(X t+s − x)
∂t∂x 2 ∂x 2
3
+ O[(|X t+s − x| + s) ]
for X t = x and y = g(t, x). Taking conditional expectations produces
∂ ∂
E(Y t+s − Y t | Y t = y)= g(t, x)s + g(t, x)µ(t, x)s
∂t ∂x
1 ∂ 2 2
+ g(t, x)σ (t, x)s + o(s).
2 ∂x 2
In similar manner,
2

∂ 2
Var(Y t+s − Y t | Y t = y)= g(t, x) σ (t, x)s + o(s).
∂x
It follows that the transformed diffusion process Y t has infinitesimal
mean and variance
∂ ∂ 1 ∂ 2 2
µ Y (t, y) = g(t, x)+ g(t, x)µ(t, x)+ g(t, x)σ (t, x)
∂t ∂x 2 ∂x 2
2
∂ 2

2
σ (t, y) = g(t, x) σ (t, x),
Y
∂x