Kolmogorov backward equations (diffusion)

The Kolmogorov backward equations, in the context of diffusion processes, are a set of partial differential equations (PDEs) that describe how the expected value of a function of the process changes over time, looking backward from a future time to the present. They are a fundamental tool in the study of stochastic processes and find applications in various fields, including finance, physics, and engineering.

More specifically, consider a diffusion process X(t), where t represents time. Let f(X(T)) be a function of the process at a fixed future time T. The Kolmogorov backward equation aims to determine the function u(x, t) = E[f(X(T)) | X(t) = x], which is the expected value of f(X(T)) given that the process is at state x at time t.

The equation is derived by considering the infinitesimal changes in the process over a small time interval and applying Ito's lemma. The resulting PDE expresses the rate of change of the expected value u(x, t) with respect to time in terms of the infinitesimal generator of the diffusion process. This generator involves derivatives with respect to the state variable x, and is typically related to the drift and diffusion coefficients that characterize the diffusion process X(t).

The general form of the Kolmogorov backward equation for a one-dimensional diffusion process is:

∂u/∂t + μ(x, t) ∂u/∂x + (1/2)σ²(x, t) ∂²u/∂x² = 0

where:

• u(x, t) is the expected value E[f(X(T)) | X(t) = x]
• μ(x, t) is the drift coefficient of the diffusion process
• σ²(x, t) is the diffusion coefficient of the diffusion process
• ∂u/∂t, ∂u/∂x, and ∂²u/∂x² are the partial derivatives of u with respect to time and space.

The equation is a backward equation because it calculates the value of u at earlier times t based on its future value at T. The final condition for the PDE is given by the function f(X(T)), i.e., u(x, T) = f(x). This equation, along with appropriate boundary conditions (depending on the specific problem), allows us to determine the function u(x, t) and hence the expected value of f(X(T)) conditional on the current state of the diffusion process.

The Kolmogorov backward equation is closely related to the Feynman-Kac formula, which provides a probabilistic representation of the solution to certain parabolic partial differential equations. Furthermore, it contrasts with the Kolmogorov forward equation (also known as the Fokker-Planck equation), which describes the evolution of the probability density function of the diffusion process forward in time. The choice between using the backward or forward equation depends on the specific problem and what is being analyzed.