- Kolmogorov backward equation
The Kolmogorov backward equation (KBE) and its adjoint the
Kolmogorov forward equation(KFE) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorovin 1931. Later it was realized that the KFE 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); 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 functioncentered 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, 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 and zero otherwise. We want to know for every state x at time t
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