Kolmogorov Forward and Backward Equations

Core Idea

The Kolmogorov equations describe how the transition density p(x,t; y,T) of a diffusion dX = μ(X)dt + σ(X)dW evolves. The backward equation ∂p/∂t + μ(x)∂p/∂x + (1/2)σ²(x)∂²p/∂x² = 0 acts on the initial variables (x,t). The forward equation (Fokker-Planck) ∂p/∂T = -(∂/∂y)[μ(y)p] + (1/2)(∂²/∂y²)[σ²(y)p] acts on the final variables (y,T). Together they characterize the full probabilistic evolution of diffusion processes.

Explainer

The Kolmogorov equations describe how the probability density of a diffusion process evolves in time. For the process dX = μ(X)dt + σ(X)dW with transition density p(x,t; y,T) = P(X(T) ∈ dy | X(t) = x)/dy, there are two complementary PDEs. The backward equation ∂p/∂t + μ(x)∂p/∂x + (1/2)σ²(x)∂²p/∂x² = 0 treats the terminal point (y,T) as fixed and differentiates with respect to the initial point (x,t). The forward equation (Fokker-Planck) ∂p/∂T = -(∂/∂y)[μ(y)p] + (1/2)(∂²/∂y²)[σ²(y)p] treats the initial point (x,t) as fixed and differentiates with respect to the terminal point (y,T).

The backward equation is a direct consequence of Itô's formula and the Feynman-Kac connection. If u(x,t) = E[g(X(T)) | X(t) = x], then u satisfies ∂u/∂t + μ(x)∂u/∂x + (1/2)σ²(x)∂²u/∂x² = 0 — this is the backward equation with g as terminal data. The transition density is the special case where g is a delta function: u(x,t) = p(x,t; y,T). The backward equation's differential operator L = μ∂/∂x + (1/2)σ²∂²/∂x² is called the infinitesimal generator of the diffusion, and it encodes how the process moves locally.

The forward equation describes how an entire distribution evolves. If at time t the process has density ρ(y,t), then ρ evolves by ∂ρ/∂t = -(∂/∂y)[μ(y)ρ] + (1/2)(∂²/∂y²)[σ²(y)ρ]. The operator on the right is the formal adjoint L* of the generator L. The two terms have clear physical meanings: -(∂/∂y)[μρ] is advection (drift carries probability in the direction of μ), and (1/2)(∂²/∂y²)[σ²ρ] is diffusion (noise spreads probability). For Brownian motion (μ=0, σ=1), the forward equation reduces to the heat equation ∂ρ/∂t = (1/2)∂²ρ/∂y² — the connection between probability diffusion and heat diffusion that Einstein exploited in his 1905 paper.

Stationary distributions are found by setting ∂ρ/∂T = 0 in the forward equation, yielding the ODE 0 = -(d/dy)[μ(y)π(y)] + (1/2)(d²/dy²)[σ²(y)π(y)]. For the Ornstein-Uhlenbeck process (μ(y) = -θy, σ = const), this gives π(y) ∝ exp(-θy²/σ²), confirming the Gaussian stationary distribution. More generally, one-dimensional diffusions with σ(y) > 0 have explicit stationary densities via the formula π(y) ∝ (1/σ²(y))exp(2∫μ(y)/σ²(y) dy), provided this integrates to a finite total. The Kolmogorov equations thus provide a complete toolkit for analyzing the transient and long-run behavior of diffusion processes.