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.
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
); 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 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.
which is equal to 1 if state x is in the target set at time s, and zero otherwise.
, the indicator function for the set B.
what is the probability of ending up in the target set at time s (sometimes called the hit probability).
serves as the final condition of the PDE, which is integrated backward in time, from s to t. Assume that the system state
, subject to the final condition
This can be derived using Itō's lemma on
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 the indicator function
: Historically, the KBE[1] was developed before the Feynman–Kac formula (1949).
With the same notation as before, the corresponding Kolmogorov forward equation is for