Markov property

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 if

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.