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Stochastic process

In the mathematics of probability, a stochastic process can be thought of as a random function. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field). Familiar examples of 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 topographies (landscapes), or composition variations of an inhomogeneous material.



A stochastic process is an indexed collection of random variables, each of which is defined on the same probability space W and takes values on the same codomain D (often the reals \R).

An important case is the discrete set

f_i: W \to D,

where i runs over some discrete index set I - for example if the probability distributions of the fi satisfy the Markov property the process is a Markov chain. fi is often called (stochastic) transition function or stochastic kernel.

In a continuous stochastic process the index set is continuous (usually space or time), resulting in an infinite number of random variables.

Each point in the sample space Ω; corresponds to a particular value for each of the random variables and the resulting function (mapping a point in the index set to the value of the random variable attached to it) is known as a realisation of the stochastic process.

A particular stochastic process is determined by specifying the joint probability distributions of the various random variables f(x).

Stochastic processes may be defined in higher dimensions by attaching a multivariate random variable to each point in the index set, which is equivalent to using a multidimensional index set. Indeed a multivariate random variable can itself be viewed as a stochastic process with index set I = \{ 1 \ldots n \}


The paradigm continuous stochastic process is that of Brownian motion. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and, since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus the random force is described by a two component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being homogeneous the force is independent of the spatial coordinates) with the domain of the two random variables being \R, giving the x and y components of the force. A treatment of Brownian motion generally also includes the effect of viscosity, resulting in an equation of motion known as the Langevin equation.

As another example, take the domain to be \N, the natural numbers, and our range to be \R, the real numbers. Then, a function f: \N \ \to \ \R is a sequence of real numbers, and a stochastic process with domain \N and range \R is a random sequence. The following questions arise:

  1. How is a random sequence specified?
  2. How do we find the answers to typical questions about sequences, such as
    1. what is the probability distribution of the value of f(i)?
    2. what is the probability that f is bounded?
    3. what is the probability that f is monotonic?
    4. what is the probability that f(i) has a limit as i \to \infty?
    5. if we construct a series from f(i), what is the probability that the series converges? What is the probability distribution of the sum?

Another important class of examples is when the domain is not a discrete space such as the natural numbers, but a continuous space such as the unit interval [0,1], the positive real numbers [0, \infty] or the entire real line, \R. In this case, we have a different set of questions that we might want to answer:

  1. How is a random function specified?
  2. How do we find the answers to typical questions about functions, such as
    1. what is the probability distribution of the value of f(x) ?
    2. what is the probability that f is bounded/integrable/continuous/differentiable...?
    3. what is the probability that f(x) has a limit as x \to \infty?
    4. what is the probability distribution of the integral \int_a^b f(x)\,dx?

There is an effective way to answer all of these questions, but it is rather technical (see Constructing Stochastic Processes below).

Interesting special cases

Constructing stochastic processes

In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.

There is at least one alternative axiomatization of probability theory by means of expectations on C-star algebras of random variables. In this case the method goes by the name of Gelfand-Naimark-Segal construction.

This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.

The Kolmogorov extension

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions f: X \to Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables [f(x1),...,f(xn)]. Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for [f(x1),...,f(xn - 1)]. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.

The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition.

Separability, or what the Kolmogorov extension does not provide

Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates [f(x1),...,f(xn)] are restricted to lie in measurable subsets of Yn. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of calculus are of this sort. For example:

  1. boundedness
  2. continuity
  3. differentiability

all require knowledge of uncountably many values of the function.

One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates {f(xi)} whose values determine the whole random function f.

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