Consider a diffusion process Xt with infinitesimal mean (t, x) and infinitesimal variance 2(t, x). If the function f(t) is strictly increasing and continuously differentiable,

Consider a diffusion process Xt with infinitesimal mean µ(t, x) and infinitesimal variance σ2(t, x). If the function f(t) is strictly increasing and continuously differentiable, then argue that Yt = Xf(t) is a diffusion process with infinitesimal mean and variance

µY (t, y) = µ[f(t), y]f

(t)

σ2 Y (t, y) = σ2[f(t), y]f

(t).

Apply this result to the situation where Yt starts at y0 and has

µY (t, y) = 0 and σ2 Y (t, y) = σ2(t). Show that Yt is normally distributed with mean and variance E(Yt) = y0 Var(Yt) =  t 0

σ2(s) ds.

(Hint: Let Xt be standard Brownian motion.)