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The mathematical space of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, -dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and ...
In 1953 the term Markov chain was used for stochastic processes with discrete or continuous index set, living on a countable or finite state space, see Doob. [1] or Chung. [2] Since the late 20th century it became more popular to consider a Markov chain as a stochastic process with discrete index set, living on a measurable state space. [3] [4] [5]
A state i is inessential if it is not essential. [2] A state is final if and only if its communicating class is closed. A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state. [1] [3]: 20
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
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.
Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is. The theorem is used in one of the standard proofs of existence of a Brownian motion , by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above.
In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times.