# Stochastic Process

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A stochastic process, $X_{t \in \mathcal{T}}$, is a random element that is a process (whose outcomes cannot be foretold, but do have some statistical properties).

## References

### 2015b

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/stochastic_process Retrieved:2015-2-16.
• In probability theory, a stochastic process, or sometimes random process (widely used) is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

In the simple case of discrete time, as opposed to continuous time, a stochastic process involves a sequence of random variables and the time series associated with these random variables (for example, see Markov chain, also known as discrete-time Markov chain). One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs; in most cases this will be the time parameter) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all take values in the same space (the codomain of the function). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.

Familiar examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks. Examples of random fields include static images, random terrain (landscapes), wind waves or composition variations of a heterogeneous material.

A generalization, the random field, is defined by letting the variables' parameters be members of a topological space instead of limited to real values representing time.

### 2009b

• (Encyclopaedia Britannica, 2009) ⇒ http://www.britannica.com/EBchecked/topic/285198/Index
• Stochastic Process: in probability theory, a process involving the operation of chance. For example, in radioactive decay every atom is subject to a fixed probability of breaking down in any given time interval. More generally, a stochastic process refers to a family of random variables indexed against some other variable or set of variables. It is one of the most general objects of study in probability. Some basic types of stochastic processes include Markov process es, Poisson processes (such as radioactive decay), and time series, with the index variable referring to time. This indexing can be either discrete or continuous, the interest being in the nature of changes of the variables with respect to time.