# Stochastic Process

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## stochastic process

[stō′kas·tik ′prä·səs]## Stochastic Process

(or random process), a process —that is, a change in the state of some system over time—whose course depends on chance and for which the probability of a particular course is defined. Brownian motion is a typical example of a stochastic process. Other examples of practical importance include turbulent flows of liquids and gases, the flow of current in an electric circuit where there occur random fluctuations in voltage and current strength (noise), and the propagation of radio waves with random fading of the radio signal as a result of changes in the propagation characteristics of the signal path. Also classified as stochastic processes are many industrial processes accompanied by random fluctuations and a number of processes encountered in geophysics (for example, changes in the earth’s magnetic field), physiology (for example, variations in the bioelectric potentials of the brain as recorded in an electroencephalogram), and economics.

In order to apply mathematical methods to the study of a stochastic process, it must be possible to represent schematically the instantaneous state of the system in the form of a point of some phase space, or state space, *R*. The stochastic process here is represented by a function *X*(*t*) of time *t* with values in *R*. The case where the points of *R* are specified by one or more numerical parameters (generalized coordinates of the system) has been best studied and is of great interest from the point of view of applications. In mathematical investigations, the term “stochastic process’ is often applied to a numerical-valued function *X*(*t*) that can assume different values depending on chance and for which a probability distribution is specified for its various possible values. Such a function is called a simple stochastic process. If the points of *R* are specified by several numerical-valued parameters, the corresponding stochastic process *X*(*t*) = {*X*_{1}(*t*), *X*_{2}(*t*), • • •, *X _{k}*(

*t*)} is said to be multiple.

The mathematical theory of stochastic processes and of more general random functions of an arbitrary argument is an important branch of probability theory. The first steps in the development of the theory of stochastic processes were made by, for example, A. A. Markov (the elder). These early efforts dealt with cases where time *t* varies discretely and the system can have only a finite number of different states. In other words, the schemes used involved sequences of dependent trials. A number of mathematicians have contributed to the development of the theory of stochastic processes that are functions of continuously varying time. These mathematicians include the Soviets E. E. Slutskii, A. N. Kolmogorov, and A. Ia. Khinchin, the Americans N. Wiener, W. Feller, and J. Doob, the Frenchman P. Lévy, and the Swede H. Cramer. The best studied areas of the theory of stochastic processes are certain special classes of processes, particularly Markov processes and stationary stochastic processes. Intensive research has also been done on a number of subclasses and generalizations of these two classes—for example, Markov chains, branching processes, processes with independent increments, martingales, and processes with stationary increments.

### REFERENCES

Markov, A. A. “Zamechatel’nyi sluchai ispytanii, sviazannykh ν tsep’.” In A. A. Markov,*Ischislenie veroiatnostei*, 4th ed. Moscow. 1924.

Slutskii, E. E.

*Izbrannye trudy*. Moscow, 1960.

Kolmogorov, A. N. “Ob analiticheskikh metodakh ν teorii veroiatnostei.”

*Uspekhi matemalicheskikh nauk*, 1938, issue 5, pp. 5–41.

Khinchin, A. Ia. “Teoriia korreliatsii statsionarnykh stokhasticheskikh protsessov.”

*Uspekhi matemalicheskikh nauk*, 1938, issue 5, pp. 42–51.

Wiener, N.

*Nelineinye zadachi ν teorii sluchainykh protsessov*. Moscow, 1961. (Translated from English.)

Doob, J.

*Veroiatnostnye protsessy*. Moscow, 1956. (Translated from English.)

Lévy, P.

*Stokhasticheskie protsessy i brounovskoe dvizhenie*. Moscow, 1972. (Translated from French.)

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A. M. IAGLOM