The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s (PDE) that arise in the theory of continuous-time continuous-state Markov process

Markov process

In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...

es. Both were published by Andrey Kolmogorov

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

 in 1931. Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new.

Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 ); we want to know the probability distribution of the state at a later time . The adjective 'forward' refers to the fact that serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly is a Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

 centered on the known initial state).

The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states B, sometimes called the target set. The target is described by a given function which is equal to 1 if state x is in the target set at time s, and zero otherwise. In other words, , the indicator function for the set B. We want to know for every state x at time what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case serves as the final condition of the PDE, which is integrated backward in time, from s to t.

Formulating the Kolmogorov backward equation

Assume that the system state evolves according to the stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

then the Kolmogorov backward equation is, using Ito's lemma

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

 on :


for , subject to the final condition .

This equation can also be derived from the Feynman-Kac formula

Feynman-Kac formula

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes. It offers a method of solving certain PDEs by simulating random paths of a stochastic process. Conversely, an important class of...

 by noting that the hit probability is the same as the expected value of over all paths that originate from state x at time t:

Historically of course the KBE (1931) was developed before the Feynman-Kac formula (1949).

Formulating the Kolmogorov forward equation

With the same notation as before, the corresponding Kolmogorov forward equation is:


for , with initial condition . For more on this equation see Fokker–Planck equation.