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In the mathematics of probability, a stochastic process is a random function. In the most common 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 processes modeled as stochastic 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.

Definition

A stochastic process is a sequence of measurable functions, that is, a random variable X defined on a probability space (Ω, S, Pr) with values in a space of functions F. The space F in turn consists of functions ID. Thus a stochastic process can also be regarded as an indexed collection of random variables {Xi}, where the index i ranges through an index set I, defined on the probability space (Ω, S, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed collection of random variables is the most common one.

A notable special case is where the index set is a discrete set I, often the nonnegative integers {0, 1, 2, 3, ...}.

In a continuous stochastic process the index set is continuous (usually space or time), resulting in an uncountably 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. In the case the index family is a real (finite or infinite) interval, the resulting function is called a sample path.

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

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 {1, ..., n}.

Examples

The paradigm continuous stochastic process is that of the Wiener process. 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.

If the index set of the process is N (the natural numbers), and the range is R (the real numbers), there are some natural questions to ask about the sample sequences of a process {Xi}iN, where a sample sequence is {X(ω)i}iN.

  1. What is the probability that each sample sequence is bounded?
  2. What is the probability that each sample sequence is monotonic?
  3. What is the probability that each sample sequence has a limit as the index approaches ∞?
  4. What is the probability that the series obtained from a sample sequence from f(i) converges?
  5. What is the probability distribution of the sum?

Similarly, if the index space I is a finite or infinite interval, we can ask about the sample paths {X(ω)t}t I

  1. What is the probability that it is bounded/integrable/continuous/differentiable...?
  2. What is the probability that it has a limit at ∞
  3. What is the probability distribution of the integral?

More Examples

Interesting special cases

Finite-dimensional distributions and law

A great deal of information about a stochastic process can often be obtained from its finite-dimensional distributions (the measures induced on the finite Cartesian product of the state space at a finite sequence of times) and law (the measure induced on the collection of all functions from the index set into the state space). For sample continuous processes, the finite-dimensional distributions determine the law, and vice versa. A suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).

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(x_1),\dots,f(x_n). Now, from this n-dimensional probability distribution we can deduce an (n − 1)-dimensional marginal probability distribution for f(x_1),\dots,f(x_{n-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(x_1), ..., f(x_n)] are restricted to lie in measurable subsets of Y_n. 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(x_i)\} whose values determine the whole random function f.

The Kolmogorov continuity theorem guarantees that processes that satisfy certain constraints on the moments of their increments are continuous.

See also

References

Papoulis, Athanasios & Pillai, S. Unnikrishna (2001). Probability, Random Variables and Stochastic Processes, McGraw-Hill Science/Engineering/Math. ISBN 0-07-281725-9.

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