Write the Kolmogorov forward equation for dX = mu(x) dt + sigma(x) dW, solve it for the stationary density in one dimension, and apply the result to dX = -kappa X dt + sigma dW.

Write the Kolmogorov forward equation for dX = mu(x) dt + sigma(x) dW, solve it for the stationary density in one dimension, and apply the result to dX = -kappa X dt + sigma dW.

Approach: Write the forward equation in flux form, set the time derivative and the flux to zero, then integrate the resulting first order equation for the density.

The Kolmogorov forward equation, also called the Fokker-Planck equation, is ∂_t p = -∂_x(mu(x) p) + (1/2) ∂_xx(sigma(x)^2 p), and its stationary density is p(x) ∝ (1/sigma(x)^2) exp(2 ∫^x mu(y)/sigma(y)^2 dy), which for mu = -kappa x and constant sigma is the N(0, sigma^2/(2 kappa)) density. Writing the equation as ∂_t p = -∂_x J with flux J = mu p - (1/2) ∂_x(sigma^2 p), a stationary density needs ∂_x J = 0, and on the whole line with an integrable density the constant must be zero, which is the zero flux condition. Solving mu p = (1/2)(sigma^2 p)' as a first order linear equation gives the stated formula. For the Ornstein-Uhlenbeck case the exponent is 2 ∫ (-kappa y / sigma^2) dy = -kappa x^2 / sigma^2, so p(x) ∝ exp(-x^2 /(2 sigma^2/(2 kappa))), confirming the variance sigma^2/(2 kappa) obtained earlier from the Ito isometry.

Follow-up: What stationary density does the same formula give for the square root diffusion, and what is its shape parameter?

Key concepts: Kolmogorov forward equation, Fokker-Planck, stationary density, zero flux.