Consider a time-homogeneous diffusion process Xt starting at x0 and having (t, x) = x+ and 2(t, x) = 2. Show that Xt is normally

Consider a time-homogeneous diffusion process Xt starting at x0 and having µ(t, x) = −αx+η and σ2(t, x) = σ2. Show that Xt is normally distributed with mean and variance E(Xt) = x0e−αt + η(1 − e−αt)

α

Var(Xt) = σ2(1 − e−2αt)

2α .

The case η = 0 and α > 0 is the Ornstein-Uhlenbeck process. (Hints:

The transformed process Yt = Xt/σ2 − η/α has infinitesimal mean

µY (t, z) = −αx/σ2 and infinitesimal variance σ2 Y (t, z) = 1.

Check Kolmogorov’s forward equation for Yt.)