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| {{More footnotes|date=November 2010}}
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| [[File:IE Real SandP Prices, Earnings, and Dividends 1871-2006.png|thumb|350px|[[Stock market]] fluctuations have been modeled by stochastic processes.]]
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| In [[probability theory]], a '''stochastic process''' {{IPAc-en|s|t|əʊ|ˈ|k|æ|s|t|ɪ|k}}, or sometimes '''random process''' (''widely used'') is a collection of [[random variable]]s; 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.
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| In the simple case of [[Discrete-time stochastic process|discrete time]], as opposed to [[Continuous-time_stochastic_process|continuous time]], a stochastic process involves a [[sequence (mathematics)|sequence]] of random variables and the [[time series]] associated with these random variables (for example, see [[Markov chain]], also known as discrete-time 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 [[function (mathematics)|function]]s 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 distribution]]s. 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 [[statistical independence|independent random variables]], in most commonly considered situations they exhibit complicated statistical correlations.
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| Familiar examples of processes modeled as stochastic time series include [[stock market]] and [[exchange rate]] fluctuations, signals such as [[Speech communication|speech]], [[sound|audio]] and [[video]], [[medicine|medical]] data such as a patient's [[Electrocardiogram|EKG]], [[Electroencephalography|EEG]], [[blood pressure]] or [[temperature]], and random movement such as [[Brownian motion]] or [[random walk]]s. Examples of random fields include static images, random [[terrain]] (landscapes), [[wind wave]]s or composition variations of a heterogeneous material.
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| == Formal definition and basic properties ==
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| === Definition ===
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| Given a [[probability space]] <math>(\Omega, \mathcal{F}, P)</math> and a [[measurable space]] <math>(S,\Sigma)</math>,
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| an ''S''-valued '''stochastic process''' is a collection of ''S''-valued
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| [[random variable]]s on <math>\Omega</math>, indexed by a [[totally ordered]] set ''T'' ("time"). That is, a stochastic process ''X'' is a collection
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| : <math> \{ X_t : t \in T \}</math>
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| where each <math>X_t</math> is an ''S''-valued random variable on <math>\Omega</math>. The space ''S'' is then called the '''state space''' of the process.
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| === Finite-dimensional distributions ===
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| Let ''X'' be an ''S''-valued stochastic process. For every finite sequence <math>T'=( t_1, \ldots, t_k ) \in T^k</math>, the ''k''-tuple <math>X_{T'} = (X_{t_1}, X_{t_2},\ldots, X_{t_k})</math> is a random variable taking values in <math>S^k</math>. The distribution <math>\mathbb{P}_{T'}(\cdot) = \mathbb{P} (X_{T'}^{-1}(\cdot))</math> of this random variable is a probability measure on <math>S^k</math>. This is called a [[finite-dimensional distribution]] of ''X''.
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| 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).
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| ==History of stochastic processes== | |
| Stochastic processes were first studied rigorously in the late 19th century to aid in understanding financial markets and [[Brownian motion]]. The first person to describe the mathematics behind Brownian motion was [[Thorvald N. Thiele]] in a paper on the method of [[least squares]] published in 1880. This was followed independently by [[Louis Bachelier]] in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. [[Albert Einstein]] (in one of his [[Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|1905 papers]]) and [[Marian Smoluchowski]] (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of [[Jean Baptiste Perrin]] in 1908.
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| An excerpt from Einstein's paper describes the fundamentals of a stochastic model:
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| <blockquote><poem>
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| "It must clearly be assumed that each individual particle executes a motion which is independent of the motions of all other particles; it will also be considered that the movements of one and the same particle in different time intervals are independent processes, as long as these time intervals are not chosen too small.
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| We introduce a time interval <math>\tau</math> into consideration, which is very small compared to the observable time intervals, but nevertheless so large that in two successive time intervals <math>\tau</math>, the motions executed by the particle can be thought of as events which are independent of each other".
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| </poem></blockquote>
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| == Construction ==
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| In the ordinary [[axiomatization]] of [[probability theory]] by means of [[measure theory]], the problem is to construct a [[sigma-algebra]] of [[measurable set|measurable subsets]] of the space of all functions, and then put a finite [[Measure (mathematics)|measure]] on it. For this purpose one traditionally uses a method called [[Kolmogorov extension theorem|Kolmogorov extension]].<ref>Karlin, Samuel & Taylor, Howard M. (1998). ''An Introduction to Stochastic Modeling'', Academic Press. ISBN 0-12-684887-4.</ref>
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| There is at least one alternative axiomatization of probability theory by means of [[expected value|expectations]] on [[C-star algebra|C-star]] [[algebra of random variables|algebras of random variables]]. In this case the method goes by the name of [[Gelfand–Naimark–Segal construction]].
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| 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.
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| === Kolmogorov extension ===
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| The [[Kolmogorov extension theorem|Kolmogorov extension]] proceeds along the following lines: assuming that a [[probability measure]] on the space of all functions <math>f: X \to Y</math> exists, then it can be used to specify the joint probability distribution of finite-dimensional random variables <math>f(x_1),\dots,f(x_n)</math>. Now, from this ''n''-dimensional probability distribution we can deduce an (''n'' − 1)-dimensional [[marginal probability distribution]] for <math>f(x_1),\dots,f(x_{n-1})</math>. 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 density function|probability densities]], the result is called the [[Chapman–Kolmogorov equation]].
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| The [[Kolmogorov extension theorem]] guarantees the existence of a stochastic process with a given family of finite-dimensional [[probability distribution]]s satisfying the Chapman–Kolmogorov compatibility condition.
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| === Separability, or what the Kolmogorov extension does not provide ===
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| Recall that in the Kolmogorov [[Axiomatic system|axiomatization]], [[measurable]] sets are the sets which have a probability or, in other words, the sets corresponding to [[yes-no question|yes/no question]]s that have a probabilistic answer.
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| The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates <math>[f(x_1), \dots , f(x_n)]</math> are restricted to lie in measurable subsets of <math>Y_n</math>. 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.
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| 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.
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| 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:
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| #[[bounded function|boundedness]]
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| #[[continuous function|continuity]]
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| #[[differentiability]]
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| all require knowledge of uncountably many values of the function.
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| One solution to this problem is to require that the stochastic process be [[separable space|separable]]. In other words, that there be some countable set of coordinates <math>\{f(x_i)\}</math> whose values determine the whole random function ''f''.
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| The [[Kolmogorov continuity theorem]] guarantees that processes that satisfy certain constraints on the [[moment (mathematics)|moments]] of their increments have continuous modifications and are therefore separable. | |
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| ==Filtrations==
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| Given a probability space <math>(\Omega,\mathcal{F},P)</math>, a '''filtration''' is a weakly increasing collection of [[sigma-algebras]] on <math>\Omega</math>, <math>\{\mathcal{F}_t, t\in T\}</math>, indexed by some totally ordered set <math>T</math>, and bounded above by <math>\mathcal{F}</math>, i.e. for ''s'',''t'' <math>\in T</math> with ''s'' < ''t'',
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| :<math>\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}</math>. | |
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| A stochastic process <math>X</math> on the same time set <math>T</math> is said to be '''adapted''' to the filtration if, for every t <math>\in T</math>, <math>X_t</math> is <math>\mathcal{F}_t</math>-measurable.<ref>Durrett, Rick. ''Probability: Theory and Examples''. Fourth Edition. Cambridge: Cambridge University Press, 2010.</ref>
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| ===The natural filtration===
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| Given a stochastic process <math>X = \{X_t : t\in T\}</math>, the '''natural filtration''' for (or induced by) this process is the filtration where <math>\mathcal{F}_t</math> is generated by all values of <math>X_s</math> up to time ''s'' = ''t'', i.e. <math>\mathcal{F}_t = \sigma(\{X_s^{-1}(A) : s\leq t,A \in S\})</math>.
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| A stochastic process is always adapted to its natural filtration.
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| == Classification== <!-- this part is still a bit of a mess -->
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| Stochastic processes can be classified according to the [[cardinality]] of its index set (usually interpreted as time) and state space.
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| === Discrete time and discrete states ===
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| If both <math>t</math> and <math>X_t</math> belong to <math>N</math>, the set of [[natural numbers]], then we have models, which lead to [[Markov chains]]. For example:
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| (a) If <math>X_t</math> means the bit (0 or 1) in position <math>t</math> of a sequence of transmitted bits, then <math>X_t</math> can be modelled as a Markov chain with two states. This leads to the error correcting [[viterbi algorithm]] in data transmission.
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| (b) If <math>X_t</math> means the combined genotype of a breeding couple in the <math>t</math>th generation in an inbreeding model, it can be shown that the proportion of heterozygous individuals in the population approaches zero as <math>t</math> goes to ∞.<ref>Allen, Linda J. S., ''An Introduction to Stochastic Processes with Applications to Biology, 2nd Edition'', Chapman and Hall, 2010, ISBN 1-4398-1882-7</ref>
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| === Continuous time and continuous state space ===
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| 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 [[wiktionary:Homogeneous|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]].<ref>Gardiner, C. ''Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences'', 3rd ed., Springer, 2004, ISBN 3540208828</ref> | |
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| === Discrete time and continuous state space ===
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| 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 {'''X'''<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>, where a sample sequence is
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| {'''X'''<sub>''i''</sub>(ω)}<sub>''i'' ∈ '''N'''</sub>.
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| # What is the [[probability]] that each sample sequence is [[bounded function|bounded]]?
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| # What is the probability that each sample sequence is [[monotonic]]?
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| # What is the probability that each sample sequence has a [[Limit of a sequence|limit]] as the index approaches ∞?
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| # What is the probability that the [[series (mathematics)|series]] obtained from a sample sequence from <math>f(i)</math> [[Convergent series|converges]]?
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| # What is the [[probability distribution]] of the sum?
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| Main applications of discrete time continuous state stochastic models include [[Markov chain Monte Carlo]] (MCMC) and the analysis of [[Time Series]].
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| === Continuous time and discrete state space ===
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| Similarly, if the index space ''I'' is a finite or infinite [[Interval (mathematics)|interval]], we can ask about the sample paths {'''X'''<sub>''t''</sub>(ω)}<sub>''t '' ∈ ''I''</sub>
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| # What is the probability that it is bounded/[[integrable]]...?
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| # What is the probability that it has a limit at ∞
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| # What is the probability distribution of the integral?
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| ==See also==
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| * [[Covariance function]]
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| * [[Dynamics of Markovian Particles|DMP]]<!-- points to disambiguation page -->
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| * [[Entropy rate]] for a stochastic process
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| * [[Ergodic process]]
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| * [[Gillespie algorithm]]
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| * [[interacting particle system]]
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| * [[Law (stochastic processes)]]
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| * [[List of stochastic processes topics]]
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| * [[Markov chain]]
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| * [[Stationary process]]
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| * [[Stochastic calculus]]
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| * [[Stochastic cellular automaton|Probabilistic cellular automaton]]
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| == References ==
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| {{Reflist}}
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| ==Further reading==
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| <div class="references-small">
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| *{{cite book | author= Wio, S. Horacio, Deza, R. Roberto & Lopez, M. Juan | title= An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics| publisher=World Scientific Publishing | year=2012 | isbn=978-981-4374-78-1}}
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| *{{cite book | author=Papoulis, Athanasios & Pillai, S. Unnikrishna | title=Probability, Random Variables and Stochastic Processes| publisher=McGraw-Hill Science/Engineering/Math | year=2001 | editor= | isbn=0-07-281725-9}}
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| *{{cite web | title=Lecture notes in ''Advanced probability theory'' | author=Boris Tsirelson | url=http://www.webcitation.org/5cfvVZ4Kd | authorlink=Boris Tsirelson}}
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| *{{cite book | first=J. L. |last=Doob | title=Stochastic Processes | publisher=Wiley | year=1953}}
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| *{{cite book | author=Klebaner, Fima C. | title=Introduction to Stochastic Calculus With Applications | publisher=Imperial College Press | year=2011| editor= | isbn=1-84816-831-4}}
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| *{{cite web | title=An Exploration of Random Processes for Engineers | author=Bruce Hajek | url=http://www.ifp.uiuc.edu/~hajek/Papers/randomprocesses.html |date=July 2006}}
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| *{{cite web | url=https://www.youtube.com/watch?v=AUSKTk9ENzg |title=An 8 foot tall Probability Machine (named Sir Francis) comparing stock market returns to the randomness of the beans dropping through the quincunx pattern|work= Index Funds Advisors [http://www.ifa.com IFA.com]}}
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| * {{cite web | url=http://www.sitmo.com/article/popular-stochastic-processes-in-finance/ |title=Popular Stochastic Processes used in Quantitative Finance|work=sitmo.com}}
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| * {{cite web | url=http://www.goldsim.com/Content.asp?PageID=455 |title=Addressing Risk and Uncertainty}}
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| <!-- Tips for referencing:
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| For websites, use the formatting below (date/year are when you accessed the web page):
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| {{cite web | title=Title of page | work=Title of Complete Work | url=http://www.example.com | date=Month Day | year=Year}}
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| For Books, use:
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| {{cite book | author=Lincoln, Abraham; Grant, U. S.; & Davis, Jefferson | title=Resolving Family Differences Peacefully | location=Gettysburg | publisher=Printing Press | year=1861 | editor=Stephen A. Douglas | isbn=0-12-345678-9}}
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| {{cite book | author=Wio, S. Horacio; Deza, Roberto R.; & Lopez, M. Juan | title= An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics (Revised edition)| location=Singapore| publisher= World Scientific Publishing| year=2012| isbn=978-981-4374-78-1}}
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| For other sources, see: [[WP:CITET]]
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| </div>
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Stochastic Process}}
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| [[Category:Stochastic processes|*]]
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| [[Category:Telecommunication theory]]
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| [[Category:Statistical models]]
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| [[Category:Statistical data types]]
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