Exercise 13.3.4 Let p(x, y; t) denote the transition probability density function of a (, ) Brownian

Exercise 13.3.4 Let p(x, y; t) denote the transition probability density function of a (μ, σ) Brownian motion starting at x; p(x, y; t) = (1/

√

2πt σ ) exp[−(y−x −

μt)2/(2σ2t) ]. Show that p satisfies Kolmogorov’s backward equation ∂p/∂t =

(σ2/2)(∂2 p/∂x2)+μ(∂p/∂x), and Kolmogorov’s forward equation (also called the Fokker–Planck equation) ∂p/∂t = (σ2/2)(∂2 p/∂y2)−μ(∂p/∂y).