Page 331 - Applied Probability
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15. Diffusion Processes
320
∂
∞
!
=
f(t, x)φ s (x, z)
zf(t, x)φ s (x, z) −
2 ∂x
−
1 ∂
2
≈ µ(t, x)f(t, x)s −
2 ∂x
1 ∂ 2 − z ∞ 2 z φ s (x, z) dzf(t, x) dz
≈ µ(t, x)f(t, x)s − σ (t, x)f(t, x) s.
2 ∂x
Using equation (15.3), one can show that these approximations are good
to order o(s). Dividing by s and sending s to 0 give the flux
∂ 1 ∂ 2
− Pr(X t ≤ x)= µ(t, x)f(t, x) − σ (t, x)f(t, x) .
∂t 2 ∂x
A final differentiation with respect to x now produces the Kolmogorov
forward equation
∂ ∂ 1 ∂ 2 2
f(t, x)= − µ(t, x)f(t, x) + σ (t, x)f(t, x) . (15.5)
∂t ∂x 2 ∂x 2
As t tends to 0, the density f(t, x) concentrates all of its mass around the
initial point x 0 .
Example 15.2.1 Standard Brownian Motion
2
If µ(t, x) = 0 and σ (t, x) = 1, then the forward equation becomes
∂ 1 ∂ 2
f(t, x) = f(t, x).
∂t 2 ∂x 2
At X 0 = 0 one can check the solution
1 x 2
f(t, x)= √ e − 2t
2πt
by straightforward differentiation. Thus, X t has a Gaussian density with
mean 0 and variance t. Here is clear that X t becomes progressively more
concentrated around its starting point as t tends to 0.
Example 15.2.2 Transformations of Standard Brownian Motion
The transformed Brownian process Y t = σX t + αt + x 0 has infinitesimal
2 2
mean and variance µ Y (t, x)= α and σ (t, x)= σ . It is clear that Y t
Y
2
is normally distributed with mean αt + x 0 and variance σ t. The further
transformation Z t = e Y t leads to a process with infinitesimal mean and
2 2
2
2
1
variance µ Z (t, z)= zα + zσ and σ (t, z)= z σ . Because Y t is normally
2 Z
distributed, Z t is lognormally distributed.