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In probability theory, a stochastic process (pronunciation: /stoʊˈkæstɪk/), or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

In the simple case of discrete time, a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same type. Type refers to the codomain of the function. Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.

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 terrain (landscapes), wind waves or composition variations of a heterogeneous material.

Formal definition and basic properties
Definition

Given a probability space \( (\Omega, \mathcal{F}, P) \) and a measurable space \((S,\mathcal{S}) \), an S-valued stochastic process is a collection of S-valued random variables on \( \Omega \), indexed by a totally ordered set T ("time"). That is, a stochastic process X is a collection

\( \{ X_t : t \in T \} \)

where each X_t is an S-valued random variable on \Omega. The space S is then called the state space of the process.
Finite-dimensional distributions

Let X be an S-valued stochastic process. For every finite subset T'=\{ t_1, \ldots, t_k \} \subseteq T, the k-tuple \( X_{T'} = (X_{t_1}, X_{t_2},\ldots, X_{t_k}) \) is a random variable taking values in S^k. The distribution \( \mathbb{P}_{T'}(\cdot) = \mathbb{P} (X_{T'}^{-1}(\cdot)) \) of this random variable is a probability measure on S^k. This is called a finite-dimensional distribution of X.

Under suitable topological restrictions, a suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).
Construction

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 [1].

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.
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 joint 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 \( f(x_1),\dots,f(x_{n-1}) \). Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the full-blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a Wiener process (in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. 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 \([f(x_1), \dots , f(x_n)] \) 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:

boundedness
continuity
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 have continuous modifications and are therefore separable.
Filtrations

Given a probability space \( (\Omega,\mathcal{F},P) \), a filtration is a weakly increasing collection of sigma-algebras on \( \Omega, \{\mathcal{F}_t, t\in T\} \), indexed by some totally ordered set T, and bounded above by \( \mathcal{F} \) . I.e. for \( s,t\in T \) with s < t,

\( \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}. \)

A stochastic process X on the same time set T is said to be adapted to the filtration if, for every t\in T, X_t is \mathcal{F}_t-measurable.[2]
The natural filtration

Given a stochastic process \( X = \{X_t : t\in T\} \), the natural filtration for (or induced by) this process is the filtration where \( \mathcal{F}_t \) is generated by all values of \( X_s \) up to time s = t. I.e. \( \mathcal{F}_t = \sigma(\{X_s^{-1}(A) : s\leq t,A\in\mathcal{S}\}) \).

A stochastic process is always adapted to its natural filtration.
Classification

Stochastic processes can be classified according to the cardinality of its index set (usually interpreted as time) and state space.
Discrete time and discrete states

If both t and \( X_t \) belong to N, the set of natural numbers, then we have models which lead to Markov chains. For example:

(a) If \( X_t \) means the bit (0 or 1) in position t of a sequence of transmitted bits, then X_t can be modelled as a Markov chain with 2 states. This leads to the error correcting viterbi algorithm in data transmission.

(b) If \( X_t \) means the combined genotype of a breeding couple in the tth generation in a inbreeding model, it can be shown that the proportion of heterozygous individuals in the population approaches zero as t goes to ∞.[3]
Continuous time and continuous state space

The paradigm of 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.[4]
Discrete time and continuous state space

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 {Xi(ω)}i ∈ N.

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

Main applications of discrete time continuous state stochastic models include Markov chain Monte Carlo (MCMC) and the analysis of Time Series.
Continuous time and discrete state space

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

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