The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov 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.
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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
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
Formulating the Kolmogorov backward equation
Assume that the system state
then the Kolmogorov backward equation is as follows
for
This equation can also be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of
Historically of course the KBE 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