Markov process

In probability theory and statistics, a Markov process, named for the Russian mathematician Andrey Markov, is a stochastic process satisfying a certain property, called the Markov property. A Markov process can be thought of as 'memoryless': loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. I.e., conditional on the present state of the system, its future and past are independent.

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. Alternately, in modelling a process, one may assume the process to be Markov, and take this as the basis for a construction. 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) state-space. Usually a Markov chain would be defined for a discrete set of times (i.e. a discrete-time Markov Chain) although some authors use the same terminology where "time" can take continuous values. Also see continuous-time Markov process.

Markov property

The general case

Let $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space with a filtration $({\mathcal {F}}_{t},\ t\in T)$, for some (totally ordered) index set $T$; and let $(S,{\mathcal {S}})$ be a measurable space. An S-valued stochastic process $X=(X_{t},\ t\in T)$ adapted to the filtration is said to possess the Markov property with respect to the $\{{\mathcal {F}}_{t}\}$ if, for each $A\in {\mathcal {S}}$ and each $s,t\in T$ with s < t,

$\mathbb {P} (X_{t}\in A|{\mathcal {F}}_{s})=\mathbb {P} (X_{t}\in A|X_{s}).$

A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration.

For discrete-time Markov chains

In the case where $S$ is a discrete set with the discrete sigma algebra and $T=\mathbb {N}$, this can be reformulated as follows:

$\mathbb {P} (X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{0}=x_{0})=\mathbb {P} (X_{n}=x_{n}|X_{n-1}=x_{n-1})$.

Examples

Gambling

See also: random walk

Suppose that you start with $10 in poker chips, and you repeatedly wager $1 on a (fair) coin toss indefinitely, or until you lose all of your poker chips. If $X_{n}$ represents the number of dollars you have in chips after n tosses, with $X_{0}=10$, then the sequence $\{X_{n}:n\in [0,\infty )\}$ is a Markov process. If I know that you have 12 chips now, then I will guess that with even odds, you will either have 11 or 13 chips after the next toss. This guess is not improved by the added knowledge that you started with 10 chips, then went up to 11, down to 10, up to 11, and then to 12.

A birth process

See also: birth-death process and Poisson process

Suppose that you are popping one hundred kernels of popcorn, and each kernel will pop at an independent, uniformly random time within the next hundred seconds. Let $X_{t}$ denote the number of kernels which have popped up to time t. Then this is a continuous time, non-homogenous Markov process. If after some amount of time, I want to guess how many kernels will pop in the next second, I need only know how many kernels have popped. It will not help me to know when they popped, so knowing $X_{t}$ for previous times t will not inform my guess.

The process described here is an approximation of a Poisson process - Poisson processes are also Markov.

A non-Markov example

Suppose that you have a coin purse containing five quarters, five nickels and five dimes, and one-by-one, you randomly draw coins from the purse and set them on a table. If $X_{n}$ represents the total value of the coins set on the table after n draws, with $X_{0}=0$, then the sequence $\{X_{n}:n\in [0,\infty )\}$ is not a Markov process.

To see why this is the case, suppose that in your first six draws, you draw all five nickels, and then a quarter. So $X_{6}=\$0.50$. If we know not just $X_{6}$, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel, so we can determine that $X_{7}\geq \$0.60$ with probability 1. But if we do not know the earlier values, then based only on the value $X_{6}$ we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about $X_{7}$ are impacted by our knowledge of values prior to $X_{6}$.

Markovian representations

In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states. For example, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X. Mathematically, this takes the form:

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

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 a non-Markovian process with a Markovian representation is a moving average time series.

See also

• Examples of Markov chains
• Semi-Markov process
• Markov decision process
• Dynamics of Markovian particles
• Random walk
• Brownian motion
• Markov chain
• Markov model

Notes

• Yosida, K. “Functional Analysis”, Ch XIII, § 3, Springer-Verlag, 1968. ISBN 3-540-58654-7
• Ribarič.M. and I.Vidav, “An inequality for concave functions.” Glasnik Matematički 8 (28), 183–186 (1973).

External links

• Weisstein, Eric W. "Markov process". MathWorld.