A stochastic (/stoʊˈkæstɪk/) process is a random process evolving with time. More specifically, in probability theory, a stochastic process is a time sequence representing the evolution of some system represented by a variable whose change is subject to a random variation. 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 many stochastic processes, the movement to the next state or position depends on only the current state, and is independent from prior states or values the process has taken. In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables. (For example, see Markov chain, also known as discrete-time Markov chain.) The random variables corresponding to various times may be completely different, the only requirement being that these different random quantities all take values in the same space (the codomain of the function). One approach may be to model these random variables as random functions of one or several deterministic arguments (in most cases, the time parameter). 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 dependence. Familiar examples of stochastic processes 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. A generalization, the random field, is defined by letting the variables be parametrized by members of a topological space instead of time. Examples of random fields include static images, random terrain (landscapes), wind waves and composition variations of a heterogeneous material.

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