1. The equations

2. Background

3. Relation with the generating function

4. History

5. See also

6. References

In the context of a continuous-time Markov process, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time-evolution of the probability , where (the state space) and are the final and initial time respectively.

The equations

For the case of countable state space we put in place of .

Kolmogorov forward equations read

while Kolmogorov backward equations are

The functions are continuous and differentiable in both time arguments. They represent the

probability that the system that was in state at time jumps to state at some later time . The continuous quantities satisfy

Background

The original derivation of the equations by Kolmogorov ^[1] starts with the Chapman-Kolmogorov equation (Kolmogorov called it Fundamental equation) for time-continuous and differentiable Markov processes on a finite, discrete state space. In this formulation, it is assumed that the probabilities are continuous and differentiable functions of . Also adequate limit properties for the derivatives are assumed. Feller ^[2] derives the equations under slightly different conditions, starting with the concept of purely discontinuous Markov process and formulating them for more general state spaces. Feller ^[2] proves the existence of solutions of probabilistic character to the Kolmogorov forward equations and Kolmogorov backward equations under natural conditions.

Relation with the generating function

Still in the discrete state case, letting and assuming that the system initially is found in state

, The Kolmogorov forward equations describe an initial value problem for finding the probabilities of the process, given the quantities . We put and

For the case of a pure death process with constant rates the only nonzero coefficients are . Letting

the system of equations can in this case be recast as a partial differential equation for with initial condition . After some manipulations, the system of equations reads,^[3]

History

A brief historical note can be found at Kolmogorov equations.

See also

• Continuous-time Markov process
• Jump process
• Master equation
• Fokker–Planck equation
• Kolmogorov backward equations (diffusion)

References

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