Markov property

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Sacara informal, prosés stokastik mibanda sipat Markov if the conditional probability distribution of future states of the process, given the present state, 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, if X(t), t > 0, is a stochastic process, the Markov property states that

\mathrm{Pr}\big[X(t+h) = y \,|\, X(s) = x(s), s \leq t\big] = \mathrm{Pr}\big[X(t+h) = y \,|\, X(t) = x(t)\big], \quad \forall h > 0.

Markov processes are typically termed (time-) homogeneous if

\mathrm{Pr}\big[X(t+h) = y \,|\, X(t) = x(t)\big] = \mathrm{Pr}\big[X(h) = y \,|\, X(0) = x(0)\big], \quad \forall t, h > 0,

and otherwise are termed (time-) inhomogeneous (or (time-) nonhomogeneous). 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. Let X be a non-Markovian process. Then we define a process Y, such that each state of Y represents a time-interval of states of X, i.e. mathematically

Y(t) = \big\{ X(s) : s \in [a(t), b(t)] \, \big\}.

If Y has the Markov property, then it is a Markovian representation of X. In this case, X is also called a second-order Markov process. Higher-order Markov processes are defined analogously.

An example of an non-Markovian process with a Markovian representation is a moving average deret waktu.

Prosés Markov nu pangkawentarna nyaéta ranté Markov, ngan prosés-prosés séjénna ogé, kaasup gerak Brown (Ing. Brownian motion), Markovian kénéh.

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