In layman's terms, a stochastic process
has the Markov property
if "the future depends only on the present, not on the past"; that is, if the probability distribution
of future states of the process depends only upon the current state, and conditionally independent of the past states (the path
of the process) given the present state. A process with the Markov property is usually called a Markov process
, and may be described as Markovian
Mathematically, the Markov property states that, if X(t), t > 0, is a stochastic process, then
Markov processes are typically termed (time-) homogeneous
and otherwise are termed (time-) inhomogeneous
. Homogeneous Markov processes, usually being simpler than inhomogeneous ones, form the most important class of Markov processes.
In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states to each include the states over an interval of times, for example
An example of an non-Markovian process with a Markovian representation is a moving average time series
The most famous Markov processes are Markov chains, but many other processes, including Brownian motion, are Markovian.
See also: Examples of Markov chains