The Azimuth Project
Random process

Contents

Idea

The theory of stochastic or random processes is the application of probability theory to situations where the random objects are functions. If one focuses on quantities that can be observed about a function, such as its pointwise values, its integrals against given test functions, its extreme values, and so on, a stochastic process is just an uncountable collection of random variables satisfying consistency conditions coming from the fact that the random variables are all observations of a function rather than a disparate collection of variables.

Random processes are a broad topic in both pure mathematical research and applications. This page is about random processes as a means of modelling in engineering and scientific applications. As such, most often the random function is a function of continuous time, time ordering and continuity play an important role, and one considers dynamical equations with random coefficients and/or driven by noise. These stochastic differential equations are the primary “consistency conditions” linking all the random variables associated with a stochastic process. Also, and especially in climate models, one may consider random fields which are a function of spatial coordinates, not time, or evolving random fields where the random function is a function of both time and space coordinates, leading to stochastic partial differential equations.

Mathematical definition

A stochastic (random) process consists of a family of random variables on common underlying probability space. If the variables form a sequence, then it is a discrete time process; if they are indexed by an interval of real numbers, it is a continuous time process.

Implementation issues

A computer model of a random process needs a random number generator. Certain random processes are solutions of stochastic differential equations and can therefore be simulated by numerical solvers of these equations.

References