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15. Diffusion Processes
at y = g(t, x).
In many cases of interest, the random variable X t has a density function
f(t, x) that depends on the initial point X 0 = x 0 . To characterize f(t, x), we
now give a heuristic derivation of Kolmogorov’s forward partial differ-
ential equation. Our approach exploits the notion of probability flux.
Here it helps to imagine a large ensemble of diffusing particles, each inde-
pendently executing the same process. We position ourselves at some point
x and record the rate at which particles pass through x from left to right
minus the rate at which they pass from right to left. This rate, normalized
by the total number of particles, is the probability flux at x. We can express
∂
the flux more formally as the negative derivative − Pr(X t ≤ x).
∂t
To calculate this time derivative, we rewrite the difference
Pr(X t ≤ x) − Pr(X t+s ≤ x)
=Pr(X t ≤ x, X t+s >x) + Pr(X t ≤ x, X t+s ≤ x)
− Pr(X t ≤ x, X t+s ≤ x) − Pr(X t >x, X t+s ≤ x)
=Pr(X t ≤ x, X t+s >x) − Pr(X t >x, X t+s ≤ x).
The first of the resulting probabilities, Pr(X t ≤ x, X t+s >x), can be
expressed as

x
∞
Pr(X t ≤ x, X t+s >x)= f(t, y)φ s (y, z) dy dz,
0 x−z
where the increment Z = X t+s − X t has density φ s (y, z) when X t = y.In
similar fashion, the second probability becomes
0 x−z

Pr(X t >x, X t+s ≤ x)= f(t, y)φ s (y, z) dy dz,
− x
producing overall

∞
x
Pr(X t ≤ x) − Pr(X t+s ≤ x)= f(t, y)φ s (y, z) dy dz.(15.4)
− x−z
Because for small values of s only values of y near x should contribute
to the flux, we substitute the first-order expansion

∂
f(t, y)φ s (y, z) ≈ f(t, x)φ s (x, z)+ f(t, x)φ s (x, z) (y − x)
∂x
in equation (15.4). In light of equations (15.1) and (15.2), this yields

Pr(X t ≤ x) − Pr(X t+s ≤ x)
∞ x ∂ !

≈ f(t, x)φ s (x, z)+ f(t, x)φ s (x, z) (y − x) dy dz
− x−z ∂x