In probability theory, Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, characterize continuous-time Markov processes. In particular, they describe how the probability that a continuous-time Markov process is in a certain state changes over time.
Diffusion processes vs. jump processes
Writing in 1931, Andrei Kolmogorov started from the theory of discrete time Markov processes, which are described by the Chapman–Kolmogorov equation, and sought to derive a theory of continuous time Markov processes by extending this equation. He found that there are two kinds of continuous time Markov processes, depending on the assumed behavior over small intervals of time:
If you assume that "in a small time interval there is an overwhelming probability that the state will remain unchanged; however, if it changes, the change may be radical", then you are led to what are called jump processes.
The other case leads to processes such as those "represented by diffusion and by Brownian motion; there it is certain that some change will occur in any time interval, however small; only, here it is certain that the changes during small time intervals will be also small".
For each of these two kinds of processes, Kolmogorov derived a forward and a backward system of equations (four in all).
William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair, in both jump and diffusion processes. Much later, in 1956, he referred to the equations for the jump process as "Kolmogorov forward equations" and "Kolmogorov backward equations".
The modern view
- In the context of a continuous-time Markov process with jumps, see Kolmogorov equations (Markov jump process). In particular, in natural sciences the forward equation is also known as master equation.
- In the context of a diffusion process, for the backward Kolmogorov equations see Kolmogorov backward equations (diffusion). The forward Kolmogorov equation is also known as Fokker–Planck equation.
An example from biology
One example from biology is given below:
This equation is applied to model population growth with birth. Where is the population index, with reference the initial population, is the birth rate, and finally , i.e. the probability of achieving a certain population size.
The analytical solution is:
This is a formula for the density in terms of the preceding ones, i.e. .
- Feller, W. (1949). "On the Theory of Stochastic Processes, with Particular Reference to Applications". Proceedings of the (First) Berkeley Symposium on Mathematical Statistics and Probability. pp. 403–432.
- Kolmogorov, Andrei (1931). "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" [On Analytical Methods in the Theory of Probability]. Mathematische Annalen (in German). 104: 415–458. doi:10.1007/BF01457949.
- Feller, William (1957). "On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations". Annals of Mathematics. 65 (3): 527–570. doi:10.2307/1970064.
- Kimura, Motoo (1957). "Some Problems of Stochastic Processes in Genetics". Annals of Mathematical Statistics. 28 (4): 882–901. JSTOR 2237051.
- Logan, J. David; Wolesensky, William R. (2009). Mathematical Methods in Biology. Pure and Applied Mathematics. John Wiley& Sons. pp. 325–327. ISBN 978-0-470-52587-6.