Stochastic Process

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 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 random function, that is a random variable X defined on a probability space (Ω , Pr) with values in a space of functions F. The space F in turn consists of functions I → D. 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, 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}i ∈ N, where a sample sequence is {X(ω)i}i ∈ N.
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?