In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a timevarying random phenomenon for which a specific property (the Markov property) holds. In a common description, a stochastic process with the Markov property, or memorylessness, is one for which conditional on the present state of the system, its future and past are independent^{[1]}.
Markov processes arise in probability and statistics in one of two ways. A stochastic process, defined via a separate argument, may be shown (mathematically) to have the Markov property and as a consequence to have the properties that can be deduced from this for all Markov processes. Of more practical importance is the use of the assumption that the Markov property holds for a certain random process in order to construct, ab initio, a stochastic model for that process. In modelling terms, assuming that the Markov property holds is one of a limited number of simple ways of introducing statistical dependence into a model for a stochastic process in such a way that allows the strength of dependence at different lags to decline as the lag increases.
Often, the term Markov chain is used to mean a Markov process which has a discrete (finite or countable) statespace. Usually a Markov chain would be defined for a discrete set of times (i.e. a discretetime Markov Chain)^{[2]} although some authors use the same terminology where "time" can take continuous values.^{[3]} Also see continuoustime Markov process.
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The Markov property
For certain types of stochastic processes it is simple to formulate the condition specifying whether the Markov property holds while, for others, more sophisticated mathematics is required as described in the article Markov property. One simple instance relates to a stochastic process whose states X can take on a discrete set of values. The states vary with time t and hence the values are denoted by X(t). The description here is the same irrespective of whether the timeindex is either a continuous variable or a discrete variable. Consider any set of "past times" ( ..., p_{2}, p_{1}), any "present time" s, and any "future time" t, where each of these times is within the range for which the stochastic process is defined, and
Then the Markov property holds, and the process is a Markov process, if the condition
holds for all sets of values ( ... ,x(p_{2}), x(p_{1}), x(s), x(t) ), and for all sets of times. The interpretation of this is that the conditional probability
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