A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state, not on the sequence of events that preceded it. A process with this property is called a Markov process. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. Both the terms "Markov property" and "strong Markov property" have been used in connection with a particular "memoryless" property of the exponential distribution.
The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model.
A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model.
A discrete-time stochastic process satisfying the Markov property is known as a Markov chain.
A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be Markovian or a Markov process. The most famous Markov process is a Markov chain. Brownian motion is another well-known Markov process.
Let be a probability space with a filtration , for some (totally ordered) index set ; and let be a measurable space. A -valued stochastic process adapted to the filtration is said to possess the Markov property if, for each and each with ,
In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows:
Alternatively, the Markov property can be formulated as follows.
for all and bounded and measurable.
Strong Markov property
Suppose that is a stochastic process on a probability space with natural filtration . Then is said to have the strong Markov property if, for each stopping time , conditioned on the event , we have that for each , is independent of given .
The strong Markov property implies the ordinary Markov property, since by taking the stopping time , the ordinary Markov property can be deduced.
- Markov chain
- Markov blanket
- Markov decision process
- Causal Markov condition
- Markov model
- Chapman–Kolmogorov equation
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