stochastic process

stochastic process (plural stochastic processes)

1. (mathematics, probability theory) A function that maps elements of an index set to elements of a collection of random variables; historically, a collection of random variables indexed by a set of numbers usually regarded as points in time.
   • 1994, Andrzej Lasota, Michael C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Springer, 2nd Edition, page 254,

     A stochastic process ${\displaystyle \{\zeta _{t}\}}$ is a family of random variables that depends on a parameter ${\displaystyle t}$ , usually called time. […] Two important properties stochastic processes may have are given in the following definition.

   • 2009, Charlotte Werndl, Deterministic Versus Indeterministic Descriptios: Not That Different After All, Alexander Hieke, Hannes Leitgeb (editors), Reduction - Abstraction - Analysis, page 71,

     Hence even Bernoulli processes, often regarded as the most random stochastic processes, are simulated by deterministic systems in science when observed with specific observation functions. But then any attempt to separate such deterministic systems from the deterministic systems needed to simulate the stochastic processes in science must fail. The philosophical implication of this result is that the above guess is wrong, viz. the deterministic systems which simulate the stochastic processes in science include deterministic systems in science. This indicates that from a predictive viewpoint, stochastic processes and deterministic systems are very similar.

   • 2014, Pierre Brémaud, Fourier Analysis and Stochastic Processes, Springer, page xi,

     A unified treatment of all the aspects of Fourier theory relevant to the Fourier analysis of stochastic processes is not only unavoidable, but also intellectually satisfying, and in fact time-saving.

A stochastic process is effectively a function of two variables: time (element of the index set) and event (point in the sample space implied by the set of random variables). Fixing the time yields a random variable. Fixing the event, on the other hand, yields a sample path.

• (function mapping to a random variable): probabilistic process, probability process, random process

• sample path