In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, in the context of
Markov processes, 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 process's distribution. This article, as opposed to the article titled
Kolmogorov equations, focuses on the scenario where we have a
continuous-time Markov chain (so the state space
is countable). In this case, we can treat the Kolmogorov equations as a way to describe the
probability , where
(the state space) and
are the final and initial times, respectively.
The equations
For the case of a
countable state space we put
in place of
.
The Kolmogorov forward equations read
:
,
where
is the
transition rate matrix (also known as the generator matrix),
while the 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 starts with the
Chapman–Kolmogorov equation In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic p ...
(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 derives the equations under slightly different conditions, starting with the concept of purely discontinuous Markov process and then formulating them for more general state spaces.
[Feller, Willy (1940) "On the Integro-Differential Equations of Purely Discontinuous Markoff Processes", ''Transactions of the American Mathematical Society'', 48 (3), 488-515 ] Feller 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 write where , then
:
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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
for with initial condition . After some manipulations, the system of equations reads,[Bailey, Norman T.J. (1990) ''The Elements of Stochastic Processes with Applications to the Natural Sciences'', Wiley. (page 90)]
:
History
A brief historical note can be found at Kolmogorov equations.
See also
* Kolmogorov equations
* Master equation (in physics and chemistry), a synonym of "Kolmogorov equations" for many continuous-time Markov chains appearing in physics and chemistry.
References
{{Reflist
Markov processes