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## Introduction

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or n-dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization.(File:Wiener process 3d.png|thumb|right|A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 â‰¤ t â‰¤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.)

### Etymology

The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence.OED, Stochastic, In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".BOOK, O. B. SheÄ­nin, Theory of probability and statistics as exemplified in short dictums,weblink 2006, NG Verlag, 978-3-938417-40-9, 5, This phrase was used, with reference to Bernoulli, by Ladislaus BortkiewiczBOOK, Oscar Sheynin, Heinrich Strecker, Alexandr A. Chuprov: Life, Work, Correspondence,weblink 2011, V&R unipress GmbH, 978-3-89971-812-6, 136, who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer ProzeÃŸ was used in German by Aleksandr Khinchin,JOURNAL, Khintchine, A., Korrelationstheorie der stationeren stochastischen Prozesse, Mathematische Annalen, 109, 1, 1934, 604â€“615, 0025-5831, 10.1007/BF01449156, though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.JOURNAL, Kolmogoroff, A., Ãœber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Mathematische Annalen, 104, 1, 1931, 1, 0025-5831, 10.1007/BF01457949, Early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.OED, Random,

### Notation

A stochastic process can be denoted, among other ways, by {X(t)}_{tin T} , {X_t}_{tin T} , {X_t},BOOK, John Lamperti, Stochastic processes: a survey of the mathematical theory,weblink 1977, Springer-Verlag, 978-3-540-90275-1, 3, {X(t)} or simply as X or X(t), although X(t) is regarded as an abuse of notation.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 55, For example, X(t) or X_t are used to refer to the random variable with the index t, and not the entire stochastic process. If the index set is T=[0,infty), then one can write, for example, (X_t , t geq 0) to denote the stochastic process.

## Examples

### Bernoulli process

One of the simplest stochastic processes is the Bernoulli process, which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p and zero with probability 1-p. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is p and its value is one, while the value of a tail is zero.BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 301, In other words, a Bernoulli process is a sequence of iid Bernoulli random variables,BOOK, Dimitri P. Bertsekas, John N. Tsitsiklis, Introduction to Probability,weblink 2002, Athena Scientific, 978-1-886529-40-3, 273, where each coin flip is an example of a Bernoulli trial.BOOK, Oliver C. Ibe, Elements of Random Walk and Diffusion Processes,weblink 29 August 2013, John Wiley & Sons, 978-1-118-61793-9, 11,

### Further examples

#### Martingale

LÃ©vy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.BOOK, Jean Bertoin, LÃ©vy Processes,weblink 29 October 1998, Cambridge University Press, 978-0-521-64632-1, viii, These processes have many applications in fields such as finance, fluid mechanics, physics and biology.JOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1336, BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, 69, The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process X is a LÃ©vy process if for n non-negatives numbers, 0leq t_1leq dots leq t_n, the corresponding n-1 incrementsX_{t_2}-X_{t_1}, dots , X_{t_{n-1}}-X_{t_n},are all independent of each other, and the distribution of each increment only depends on the difference in time.A LÃ©vy process can be defined such that its state space is some abstract mathematical space, such as a Banach space, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so I= [0,infty) , which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and subordinators are all LÃ©vy processes.

#### Random field

A random field is a collection of random variables indexed by a n-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions.BOOK, Leonid Koralov, Yakov G. Sinai, Theory of Probability and Random Processes,weblink 10 August 2007, Springer Science & Business Media, 978-3-540-68829-7, 171, If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, 19,

## Definitions

### Stochastic process

A stochastic process is defined as a collection of random variables defined on a common probability space (Omega, mathcal{F}, P), where Omega is a sample space, mathcal{F} is a sigma-algebra, and P is a probability measure, and the random variables, indexed by some set T, all take values in the same mathematical space S, which must be measurable with respect to some sigma-algebra Sigma.In other words, for a given probability space (Omega, mathcal{F}, P) and a measurable space (S,Sigma), a stochastic process is a collection of S-valued random variables, which can be written as:BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 293, {X(t):tin T }.Historically, in many problems from the natural sciences a point tin T had the meaning of time, so X(t) is a random variable representing a value observed at time t.BOOK, Alexander A. Borovkov, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 528, A stochastic process can also be written as {X(t,omega):tin T } to reflect that it is actually a function of two variables, tin T and omegain Omega.BOOK, Georg Lindgren, Holger Rootzen, Maria Sandsten, Stationary Stochastic Processes for Scientists and Engineers,weblink 11 October 2013, CRC Press, 978-1-4665-8618-5, 11, There are others ways to consider a stochastic process, with the above definition being considered the traditional one.BOOK, L. C. G. Rogers, David Williams, Diffusions, Markov Processes, and Martingales: Volume 1, Foundations,weblink 13 April 2000, Cambridge University Press, 978-1-107-71749-7, 121 and 122, BOOK, SÃ¸ren Asmussen, Applied Probability and Queues,weblink 15 May 2003, Springer Science & Business Media, 978-0-387-00211-8, 408, For example, a stochastic process can be interpreted or defined as a S^T-valued random variable, where S^T is the space of all the possible S-valued functions of tin T that map from the set T into the space S.

### Index set

The set T is called the index set or parameter setBOOK, Valeriy Skorokhod, Basic Principles and Applications of Probability Theory,weblink 5 December 2005, Springer Science & Business Media, 978-3-540-26312-8, 93 and 94, of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set T the interpretation of time. In addition to these sets, the index set T can be other linearly ordered sets or more general mathematical sets,BOOK, Patrick Billingsley, Probability and Measure,weblink 4 August 2008, Wiley India Pvt. Limited, 978-81-265-1771-8, 482, such as the Cartesian plane R^2 or n-dimensional Euclidean space, where an element tin T can represent a point in space.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 27, BOOK, Donald L. Snyder, Michael I. Miller, Random Point Processes in Time and Space,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-3166-0, 25, But in general more results and theorems are possible for stochastic processes when the index set is ordered.BOOK, Valeriy Skorokhod, Basic Principles and Applications of Probability Theory,weblink 5 December 2005, Springer Science & Business Media, 978-3-540-26312-8, 104,

### State space

The mathematical space S of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, n-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take. BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 294 and 295, BOOK, Pierre BrÃ©maud, Fourier Analysis and Stochastic Processes,weblink 16 September 2014, Springer, 978-3-319-09590-5, 120,

### Sample function

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 296, More precisely, if {X(t,omega):tin T } is a stochastic process, then for any point omegainOmega, the mappingX(cdot,omega): T rightarrow S,is called a sample function, a realization, or, particularly when T is interpreted as time, a sample path of the stochastic process {X(t,omega):tin T }.BOOK, L. C. G. Rogers, David Williams, Diffusions, Markov Processes, and Martingales: Volume 1, Foundations,weblink 13 April 2000, Cambridge University Press, 978-1-107-71749-7, 121â€“124, This means that for a fixed omegainOmega, there exists a sample function that maps the index set T to the state space S. Other names for a sample function of a stochastic process include trajectory, path functionBOOK, Patrick Billingsley, Probability and Measure,weblink 4 August 2008, Wiley India Pvt. Limited, 978-81-265-1771-8, 493, or path.BOOK, Bernt Ã˜ksendal, Stochastic Differential Equations: An Introduction with Applications,weblink 2003, Springer Science & Business Media, 978-3-540-04758-2, 10,

### Increment

An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if {X(t):tin T } is a stochastic process with state space S and index set T=[0,infty), then for any two non-negative numbers t_1in [0,infty) and t_2in [0,infty) such that t_1leq t_2, the difference X_{t_2}-X_{t_1} is a S-valued random variable known as an increment. When interested in the increments, often the state space S is the real line or the natural numbers, but it can be n-dimensional Euclidean space or more abstract spaces such as Banach spaces.

### Further definitions

#### Law

For a stochastic process XcolonOmega rightarrow S^T defined on the probability space (Omega, mathcal{F}, P), the law of stochastic process X is defined as the image measure:mu=Pcirc X^{-1},where P is a probability measure, the symbol circ denotes function composition and X^{-1} is the pre-image of the measurable function or, equivalently, the S^T-valued random variable X, where S^T is the space of all the possible S-valued functions of tin T, so the law of a stochastic process is a probability measure.BOOK, Sidney I. Resnick, Adventures in Stochastic Processes,weblink 11 December 2013, Springer Science & Business Media, 978-1-4612-0387-2, 40â€“41, For a measurable subset B of S^T, the pre-image of X givesX^{-1}(B)={omegain Omega: X(omega)in B },so the law of a X can be written as:mu(B)=P({omegain Omega: X(omega)in B }).The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution.BOOK, Ward Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues,weblink 11 April 2006, Springer Science & Business Media, 978-0-387-21748-2, 23, BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, 4, BOOK, Daniel Revuz, Marc Yor, Continuous Martingales and Brownian Motion,weblink 9 March 2013, Springer Science & Business Media, 978-3-662-06400-9, 10,

#### Finite-dimensional probability distributions

For a stochastic process X with law mu, its finite-dimensional distributions are defined as:mu_{t_1,dots,t_n} =Pcirc (X({t_1}),dots,X({t_n}))^{-1},where ngeq 1 is a counting number and each set t_i is a non-empty finite subset of the index set T, so each t_isubset T, which means that t_1,dots,t_n is any finite collection of subsets of the index set T.BOOK, L. C. G. Rogers, David Williams, Diffusions, Markov Processes, and Martingales: Volume 1, Foundations,weblink 13 April 2000, Cambridge University Press, 978-1-107-71749-7, 123, For any measurable subset C of the n-fold Cartesian power S^n=Stimesdots times S, the finite-dimensional distributions of a stochastic process X can be written as:mu_{t_1,dots,t_n}(C) =P Big(big{omegain Omega: big( X_{t_1}(omega), dots, X_{t_n}(omega) big) in C big} Big).The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.

#### Filtration

A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration {mathcal{F}_t}_{tin T} , on a probability space (Omega, mathcal{F}, P) is a family of sigma-algebras such that mathcal{F}_s subseteq mathcal{F}_t subseteq mathcal{F} for all s leq t, where t, sin T and leq denotes the total order of the index set T. With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process X_t at tin T, which can be interpreted as time t. The intuition behind a filtration mathcal{F}_t is that as time t passes, more and more information on X_t is known or available, which is captured in mathcal{F}_t, resulting in finer and finer partitions of Omega.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 22â€“23, BOOK, Peter MÃ¶rters, Yuval Peres, Brownian Motion,weblink 25 March 2010, Cambridge University Press, 978-1-139-48657-6, 37,

#### Indistinguishable

Two stochastic processes X and Y defined on the same probability space (Omega,mathcal{F},P) with the same index set T and set space S are said be indistinguishable if the followingP(X_t=Y_t text{ for all } tin T )=1 ,holds. If two X and Y are modifications of each other and are almost surely continuous, then X and Y are indistinguishable.BOOK, Monique Jeanblanc, Marc Yor, Marc Chesney, Mathematical Methods for Financial Markets,weblink 13 October 2009, Springer Science & Business Media, 978-1-85233-376-8, 11,

#### Independence

Two stochastic processes X and Y defined on the same probability space (Omega,mathcal{F},P) with the same index set T are said be independent if for all n in mathbb{N} and for every choice of epochs t_1,ldots,t_n in T, the random vectors left( X(t_1),ldots,X(t_n) right) and left( Y(t_1),ldots,Y(t_n) right) are independent.Lapidoth, Amos, A Foundation in Digital Communication, Cambridge University Press, 2009.{{rp|p. 515}}

#### Uncorrelatedness

Two stochastic processes left{X_tright} and left{Y_tright} are called uncorrelated if their cross-covariance operatorname{K}_{mathbf{X}mathbf{Y}}(t_1,t_2) = operatorname{E} left[ left( X(t_1)- mu_X(t_1) right) left( Y(t_2)- mu_Y(t_2) right) right] is zero for all times.Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3{{rp|p. 142}} Formally:

#### Independence implies uncorrelatedness

If two stochastic processes X and Y are independent, then they are also uncorrelated.{{rp|p. 151}}

#### Orthogonality

Two stochastic processes left{X_tright} and left{Y_tright} are called orthogonal if their cross-correlation operatorname{R}_{mathbf{X}mathbf{Y}}(t_1,t_2) = operatorname{E}[X(t_1) overline{Y(t_2)}] is zero for all times.{{rp|p. 142}} Formally:

#### Regularity

In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 532, BOOK, Davar Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields,weblink 10 April 2006, Springer Science & Business Media, 978-0-387-21631-7, 148â€“165, For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.BOOK, Petar Todorovic, An Introduction to Stochastic Processes and Their Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4613-9742-7, 22, BOOK, Ward Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues,weblink 11 April 2006, Springer Science & Business Media, 978-0-387-21748-2, 79,

## History

### Early probability theory

Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,BOOK, Markov Chains: From Theory to Implementation and Experimentation, Gagniuc, Paul A., John Wiley & Sons, 2017, 978-1-119-38755-8, USA, NJ, 1â€“2, JOURNAL, David, F. N., Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability), Biometrika, 42, 1/2, 1â€“15, 1955, 0006-3444, 10.2307/2333419, 2333419, but very little analysis on them was done in terms of probability.BOOK, L. E. Maistrov, Probability Theory: A Historical Sketch,weblink 3 July 2014, Elsevier Science, 978-1-4832-1863-2, 1, The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem.BOOK, Seneta, E., Encyclopedia of Statistical Sciences, Probability, History of, 2006, 10.1002/0471667196.ess2065.pub2, 1, 978-0471667193, BOOK, John Tabak, Probability and Statistics: The Science of Uncertainty,weblink 14 May 2014, Infobase Publishing, 978-0-8160-6873-9, 24â€“26, But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.JOURNAL, Bellhouse, David, Decoding Cardano's Liber de Ludo Aleae, Historia Mathematica, 32, 2, 2005, 180â€“202, 0315-0860, 10.1016/j.hm.2004.04.001, After Cardano, Jakob Bernoulli{{efn|Also known as James or Jacques Bernoulli.BOOK, Anders Hald, A History of Probability and Statistics and Their Applications before 1750,weblink 25 February 2005, John Wiley & Sons, 978-0-471-72517-6, 221, }} wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.BOOK, L. E. Maistrov, Probability Theory: A Historical Sketch,weblink 3 July 2014, Elsevier Science, 978-1-4832-1863-2, 56, BOOK, John Tabak, Probability and Statistics: The Science of Uncertainty,weblink 14 May 2014, Infobase Publishing, 978-0-8160-6873-9, 37, But despite some renown mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, SimÃ©on Poisson and Pafnuty Chebyshev,JOURNAL, Chung, Kai Lai, Probability and Doob, The American Mathematical Monthly, 105, 1, 28â€“35, 1998, 0002-9890, 10.2307/2589523, 2589523, JOURNAL, Bingham, N., Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov, Biometrika, 87, 1, 2000, 145â€“156, 0006-3444, 10.1093/biomet/87.1.145, most of the mathematical community{{efn|It has been remarked that a notable exception was the St Petersburg School in Russia, where mathematicians led by Chebyshev studied probability theory.JOURNAL, Benzi, Margherita, Benzi, Michele, Seneta, Eugene, Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966, International Statistical Review, 75, 2, 2007, 128, 0306-7734, 10.1111/j.1751-5823.2007.00009.x, }} did not consider probability theory to be part of mathematics until the 20th century.JOURNAL, Doob, Joseph L., The Development of Rigor in Mathematical Probability (1900-1950), The American Mathematical Monthly, 103, 7, 586â€“595, 1996, 0002-9890, 10.2307/2974673, 2974673, JOURNAL, Cramer, Harald, Half a Century with Probability Theory: Some Personal Recollections, The Annals of Probability, 4, 4, 1976, 509â€“546, 0091-1798, 10.1214/aop/1176996025,

### Statistical mechanics

In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.JOURNAL, Truesdell, C., Early kinetic theories of gases, Archive for History of Exact Sciences, 15, 1, 1975, 22â€“23, 0003-9519, 10.1007/BF00327232, JOURNAL, Brush, Stephen G., Foundations of statistical mechanics 1845?1915, Archive for History of Exact Sciences, 4, 3, 1967, 150â€“151, 0003-9519, 10.1007/BF00412958, This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.JOURNAL, Truesdell, C., Early kinetic theories of gases, Archive for History of Exact Sciences, 15, 1, 1975, 31â€“32, 0003-9519, 10.1007/BF00327232, JOURNAL, Brush, S.G., The development of the kinetic theory of gases IV. Maxwell, Annals of Science, 14, 4, 1958, 243â€“255, 0003-3790, 10.1080/00033795800200147, The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.JOURNAL, Brush, Stephen G., A history of random processes, Archive for History of Exact Sciences, 5, 1, 1968, 15â€“16, 0003-9519, 10.1007/BF00328110,

### Measure theory and probability theory

In 1900 at the International Congress of Mathematicians in Paris David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms. Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, Henri Lebesgue and Ã‰mile Borel. In 1925 another French mathematician Paul LÃ©vy published the first probability book that used ideas from measure theory.In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein, Aleksandr Khinchin,{{efn|The name Khinchin is also written in (or transliterated into) English as Khintchine.JOURNAL, Doob, Joseph, Stochastic Processes and Statistics, Proceedings of the National Academy of Sciences of the United States of America, 20, 6, 1934, 376â€“379, 10.1073/pnas.20.6.376, 16587907, 1076423, 1934PNAS...20..376D, }} and Andrei Kolmogorov. Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.JOURNAL, Kendall, D. G., Batchelor, G. K., Bingham, N. H., Hayman, W. K., Hyland, J. M. E., Lorentz, G. G., Moffatt, H. K., Parry, W., Razborov, A. A., Robinson, C. A., Whittle, P., Andrei Nikolaevich Kolmogorov (1903â€“1987), Bulletin of the London Mathematical Society, 22, 1, 1990, 33, 0024-6093, 10.1112/blms/22.1.31, In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov,BOOK, Vere-Jones, David, Encyclopedia of Statistical Sciences, Khinchin, Aleksandr Yakovlevich, 1, 2006, 10.1002/0471667196.ess6027.pub2, 978-0471667193, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.BOOK, Vere-Jones, David, Encyclopedia of Statistical Sciences, Khinchin, Aleksandr Yakovlevich, 4, 2006, 10.1002/0471667196.ess6027.pub2, 978-0471667193, {{efn|Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.JOURNAL, Snell, J. Laurie, Obituary: Joseph Leonard Doob, Journal of Applied Probability, 42, 1, 2005, 251, 0021-9002, 10.1239/jap/1110381384, }}

### Birth of modern probability theory

In 1933 Andrei Kolmogorov published in German his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,{{efn|Later translated into English and published in 1950 as Foundations of the Theory of Probability}} where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as Joseph Doob, William Feller, Maurice FrÃ©chet, Paul LÃ©vy, Wolfgang Doeblin, and Harald CramÃ©r.Decades later CramÃ©r referred to the 1930s as the "heroic period of mathematical probability theory". World War II greatly interrupted the development of probability theory, causing, for example, the migration of Feller from Sweden to the United States of America and the death of Doeblin, considered now a pioneer in stochastic processes.JOURNAL, Lindvall, Torgny, W. Doeblin, 1915-1940, The Annals of Probability, 19, 3, 1991, 929â€“934, 0091-1798, 10.1214/aop/1176990329, (File:Joseph Doob.jpg|thumb|right|Mathematician Joseph Doob did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales. His book Stochastic Processes is considered highly influential in the field of probability theory. )

### Discoveries of specific stochastic processes

Although Khinchin gave mathematical definitions of stochastic processes in the 1930s, specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process. Some families of stochastic processes such as point processes or renewal processes have long and complex histories, stretching back centuries.BOOK, D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods,weblink 10 April 2006, Springer Science & Business Media, 978-0-387-21564-8, 1â€“4,

#### Bernoulli process

The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. The process is a sequence of independent Bernoulli trials, which are named after Jackob Bernoulli who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens.BOOK, Anders Hald, A History of Probability and Statistics and Their Applications before 1750,weblink 25 February 2005, John Wiley & Sons, 978-0-471-72517-6, 226, Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi in 1713.BOOK, Joel Louis Lebowitz, Nonequilibrium phenomena II: from stochastics to hydrodynamics,weblink 1984, North-Holland Pub., 978-0-444-86806-0, 8â€“10,

#### Wiener process

The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. In 1880, Thorvald Thiele wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis.JOURNAL, Hald, A., T. N. Thiele's Contributions to Statistics, International Statistical Review / Revue Internationale de Statistique, 49, 1, 1981, 1â€“20, 0306-7734, 10.2307/1403034, 1403034, JOURNAL, Lauritzen, Steffen L., Time Series Analysis in 1880: A Discussion of Contributions Made by T.N. Thiele, International Statistical Review / Revue Internationale de Statistique, 49, 3, 1981, 319â€“320, 0306-7734, 10.2307/1402616, 1402616, The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.File:Wiener Zurich1932.tif|thumb|200px|Norbert Wiener gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of Thorvald Thiele, Louis Bachelier, and Albert EinsteinAlbert EinsteinThe French mathematician Louis Bachelier used a Wiener process in his 1900 thesis in order to model price changes on the Paris Bourse, a stock exchange,JOURNAL, Courtault, Jean-Michel, Kabanov, Yuri, Bru, Bernard, Crepel, Pierre, Lebon, Isabelle, Le Marchand, Arnaud, Louis Bachelier on the Centenary of Theorie de la Speculation, Mathematical Finance, 10, 3, 2000, 339â€“353, 0960-1627, 10.1111/1467-9965.00098, without knowing the work of Thiele. It has been speculated that Bachelier drew ideas from the random walk model of Jules Regnault, but Bachelier did not cite him,JOURNAL, Jovanovic, Franck, Bachelier: Not the forgotten forerunner he has been depicted as. An analysis of the dissemination of Louis Bachelier's work in economics, The European Journal of the History of Economic Thought, 19, 3, 2012, 431â€“451, 0967-2567, 10.1080/09672567.2010.540343, and Bachelier's thesis is now considered pioneering in the field of financial mathematics.It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the Leonard Savage, and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas, which was cited by mathematicians including Doob, Feller and Kolmogorov. The book continued to be cited, but then starting in the 1960s the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.In 1905 Albert Einstein published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the kinetic theory of gases. Einstein derived a differential equation, known as a diffusion equation, for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, Marian Smoluchowski published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.JOURNAL, Brush, Stephen G., A history of random processes, Archive for History of Exact Sciences, 5, 1, 1968, 25, 0003-9519, 10.1007/BF00328110, Einstein's work, as well as experimental results obtained by Jean Perrin, later inspired Norbert Wiener in the 1920sJOURNAL, Brush, Stephen G., A history of random processes, Archive for History of Exact Sciences, 5, 1, 1968, 1â€“36, 0003-9519, 10.1007/BF00328110, to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object.

#### Poisson process

The Poisson process is named after SimÃ©on Poisson, due to its definition involving the Poisson distribution, but Poisson never studied the process.BOOK, D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods,weblink 10 April 2006, Springer Science & Business Media, 978-0-387-21564-8, 8â€“9, There are a number of claims for early uses or discoveries of the Poissonprocess.At the beginning of the 20th century the Poisson process would arise independently in different situations.In Sweden 1903, Filip Lundberg published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.BOOK, Embrechts, Paul, Stochastic Processes: Theory and Methods, Frey, RÃ¼diger, Furrer, HansjÃ¶rg, Stochastic processes in insurance and finance, 19, 2001, 367, 0169-7161, 10.1016/S0169-7161(01)19014-0, Handbook of Statistics, 9780444500144, JOURNAL, CramÃ©r, Harald, Historical review of Filip Lundberg's works on risk theory, Scandinavian Actuarial Journal, 1969, sup3, 1969, 6â€“12, 0346-1238, 10.1080/03461238.1969.10404602, Another discovery occurred in Denmark in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.In 1910 Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles. Motivated by their work, Harry Bateman studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process. After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.

#### Markov processes

LÃ©vy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul LÃ©vy who started studying them in the 1930s, but they have connections to infinitely divisible distributions going back to the 1920s. In a 1932 paper Kolmogorov derived a characteristic function for random variables associated with LÃ©vy processes. This result was later derived under more general conditions by LÃ©vy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937.BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, 67, In addition to LÃ©vy, Khinchin and Kolomogrov, early fundamental contributions to the theory of LÃ©vy processes were made by Bruno de Finetti and Kiyosi ItÃ´.

## Mathematical construction

In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically. There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.BOOK, Robert J. Adler, The Geometry of Random Fields,weblink 28 January 2010, SIAM, 978-0-89871-693-1, 13, Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem{{efn|The theorem has other names including Kolmogorov's consistency theorem,BOOK, Krishna B. Athreya, Soumendra N. Lahiri, Measure Theory and Probability Theory,weblink 27 July 2006, Springer Science & Business Media, 978-0-387-32903-1, Kolmogorov's extension theoremBOOK, Bernt Ã˜ksendal, Stochastic Differential Equations: An Introduction with Applications,weblink 2003, Springer Science & Business Media, 978-3-540-04758-2, 11, or the Daniellâ€“Kolmogorov theorem.BOOK, David Williams, Probability with Martingales,weblink 14 February 1991, Cambridge University Press, 978-0-521-40605-5, 124, }} to prove a corresponding stochastic process exists. This theorem, which is an existence theorem for measures on infinite product spaces,BOOK, Rick Durrett, Probability: Theory and Examples,weblink 30 August 2010, Cambridge University Press, 978-1-139-49113-6, 410, says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions.

### Construction issues

When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes. One problem is that is it possible to have more than one stochastic process with the same finite-dimensional distributions. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions.BOOK, Patrick Billingsley, Probability and Measure,weblink 4 August 2008, Wiley India Pvt. Limited, 978-81-265-1771-8, 493â€“494, This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 529â€“530, Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable. For a continuous-time stochastic process X, other characteristics that depend on an uncountable number of points of the index set T include:
To overcome these two difficulties, different assumptions and approaches are possible.

### Resolving construction issues

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## References

{{Reflist}}

{{further cleanup|date=July 2018}}

### Articles

• JOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1336â€“1347,
• JOURNAL, Cramer, Harald, Half a Century with Probability Theory: Some Personal Recollections, The Annals of Probability, 4, 4, 1976, 509â€“546, 0091-1798, 10.1214/aop/1176996025,
• JOURNAL, Guttorp, Peter, Thorarinsdottir, Thordis L., What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes, International Statistical Review, 80, 2, 2012, 253â€“268, 0306-7734, 10.1111/j.1751-5823.2012.00181.x,
• BOOK, Jarrow, Robert, A Festschrift for Herman Rubin, Protter, Philip, A short history of stochastic integration and mathematical finance: the early years, 1880â€“1970, 2004, 75â€“91, 0749-2170, 10.1214/lnms/1196285381, Institute of Mathematical Statistics Lecture Notes - Monograph Series, 978-0-940600-61-4,
• JOURNAL, Meyer, Paul-AndrÃ©, Stochastic Processes from 1950 to the Present, Electronic Journal for History of Probability and Statistics, 5, 1, 2009, 1â€“42,

### Books

• BOOK, Robert J. Adler, The Geometry of Random Fields,weblink SIAM, 978-0-89871-693-1, 2010-01-28,
• BOOK, Robert J. Adler, Jonathan E. Taylor, Random Fields and Geometry,weblink Springer Science & Business Media, 978-0-387-48116-6, 2009-01-29,
• BOOK, Pierre BrÃ©maud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,weblink Springer Science & Business Media, 978-1-4757-3124-8, 2013-03-09,
• BOOK, Joseph L. Doob, Stochastic processes,weblink Wiley, 1990,
• BOOK, Anders Hald, A History of Probability and Statistics and Their Applications before 1750,weblink John Wiley & Sons, 978-0-471-72517-6, 2005-02-25,
• BOOK, Crispin Gardiner,weblink Stochastic Methods, Springer, 978-3-540-70712-7, 2010-10-19,
• BOOK, Iosif I. Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink Courier Corporation, 978-0-486-69387-3, 1996,
• BOOK, Emanuel Parzen, Stochastic Processes,weblink Courier Dover Publications, 978-0-486-79688-8, 2015-06-17,
• BOOK, Murray Rosenblatt, Random Processes,weblink Oxford University Press, 1962,

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