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stochastic process

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**stochastic**or

**random process**is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.BOOK, Joseph L. Doob, Stochastipoic processes,weblink 1990, Wiley, 46 and 47, BOOK, Emanuel Parzen, Stochastic Processes,weblink 17 June 2015, Courier Dover Publications, 978-0-486-79688-8, 7 and 8, BOOK, Iosif Ilyich Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, 1, BOOK, Markov Chains: From Theory to Implementation and Experimentation, Gagniuc, Paul A., John Wiley & Sons, 2017, 978-1-119-38755-8, USA, NJ, 1â€“235, Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology,BOOK, Paul C. Bressloff, Stochastic Processes in Cell Biology,weblink 22 August 2014, Springer, 978-3-319-08488-6, chemistry,BOOK, N.G. Van Kampen, Stochastic Processes in Physics and Chemistry,weblink 30 August 2011, Elsevier, 978-0-08-047536-3, ecology,BOOK, Russell Lande, Steinar Engen, Bernt-Erik SÃ¦ther, Stochastic Population Dynamics in Ecology and Conservation,weblink 2003, Oxford University Press, 978-0-19-852525-7, neuroscience,BOOK, Carlo Laing, Gabriel J Lord, Stochastic Methods in Neuroscience,weblink 2010, OUP Oxford, 978-0-19-923507-0, and physicsBOOK, Wolfgang Paul, JÃ¶rg Baschnagel, Stochastic Processes: From Physics to Finance,weblink 11 July 2013, Springer Science & Business Media, 978-3-319-00327-6, as well as technology and engineering fields such as image processing, signal processing,BOOK, Edward R. Dougherty, Random processes for image and signal processing,weblink 1999, SPIE Optical Engineering Press, 978-0-8194-2513-3, information theory,BOOK, Thomas M. Cover, Joy A. Thomas, Elements of Information Theory,weblink 28 November 2012, John Wiley & Sons, 978-1-118-58577-1, 71, computer science,BOOK, Michael Baron, Probability and Statistics for Computer Scientists, Second Edition,weblink 15 September 2015, CRC Press, 978-1-4987-6060-7, 131, cryptographyBOOK, Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography: Principles and Protocols,weblink 2007-08-31, CRC Press, 978-1-58488-586-3, 26, and telecommunications.BOOK, FranÃ§ois Baccelli, Bartlomiej Blaszczyszyn, Stochastic Geometry and Wireless Networks,weblink 2009, Now Publishers Inc, 978-1-60198-264-3, Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 2001, Springer Science & Business Media, 978-0-387-95016-7, BOOK, Marek Musiela, Marek Rutkowski, Martingale Methods in Financial Modelling,weblink 21 January 2006, Springer Science & Business Media, 978-3-540-26653-2, BOOK, Steven E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models,weblink 3 June 2004, Springer Science & Business Media, 978-0-387-40101-0, Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process,{{efn|The term

*Brownian motion*can refer to the physical process, also known as

*Brownian movement*, and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms

*Brownian motion process*or

*Wiener process*for the latter in a style similar to, for example, Gikhman and SkorokhodBOOK, Iosif Ilyich Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, or Rosenblatt.BOOK, Murray Rosenblatt, Random Processes,weblink 1962, Oxford University Press, }} used by Louis Bachelier to study price changes on the Paris Bourse,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€“80, 0749-2170, 10.1214/lnms/1196285381, 10.1.1.114.632, Institute of Mathematical Statistics Lecture Notes - Monograph Series, 978-0-940600-61-4, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.JOURNAL, Stirzaker, David, Advice to Hedgehogs, or, Constants Can Vary, The Mathematical Gazette, 84, 500, 2000, 197â€“210, 0025-5572, 10.2307/3621649, 3621649, These two stochastic processes are considered the most important and central in the theory of stochastic processes,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, 32, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.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, The term

**random function**is also used to refer to a stochastic or random process,BOOK, Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory,weblink 10 July 2010, Springer Science & Business Media, 978-0-387-87862-1, 21, BOOK, Valeriy Skorokhod, Basic Principles and Applications of Probability Theory,weblink 5 December 2005, Springer Science & Business Media, 978-3-540-26312-8, 42, because a stochastic process can also be interpreted as a random element in a function space.BOOK, John Lamperti, Stochastic processes: a survey of the mathematical theory,weblink 1977, Springer-Verlag, 978-3-540-90275-1, 1â€“2, The terms

*stochastic process*and

*random process*are used interchangeably, often with no specific mathematical space for the set that indexes the random variables.BOOK, Olav Kallenberg, Foundations of Modern Probability,weblink 8 January 2002, Springer Science & Business Media, 978-0-387-95313-7, 24â€“25, BOOK, LoÃ¯c Chaumont, Marc Yor, Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning,weblink 19 July 2012, Cambridge University Press, 978-1-107-60655-5, 175, But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.BOOK, Robert J. Adler, Jonathan E. Taylor, Random Fields and Geometry,weblink 29 January 2009, Springer Science & Business Media, 978-0-387-48116-6, 7â€“8, The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks,BOOK, Gregory F. Lawler, Vlada Limic, Random Walk: A Modern Introduction,weblink 24 June 2010, Cambridge University Press, 978-1-139-48876-1, martingales,BOOK, David Williams, Probability with Martingales,weblink 14 February 1991, Cambridge University Press, 978-0-521-40605-5, Markov processes,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, LÃ©vy processes,BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, Gaussian processes,BOOK, Mikhail Lifshits, Lectures on Gaussian Processes,weblink 2012-01-11, Springer Science & Business Media, 978-3-642-24939-6, random fields,BOOK, Robert J. Adler, The Geometry of Random Fields,weblink 28 January 2010, SIAM, 978-0-89871-693-1, renewal processes, and branching processes.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topologyBOOK, Bruce Hajek, Random Processes for Engineers,weblink 12 March 2015, Cambridge University Press, 978-1-316-24124-0, BOOK, G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling,weblink 1 January 1999, SIAM, 978-0-89871-425-8, BOOK, D.J. Daley, David Vere-Jones, An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure,weblink 12 November 2007, Springer Science & Business Media, 978-0-387-21337-8, as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.BOOK, Patrick Billingsley, Probability and Measure,weblink 4 August 2008, Wiley India Pvt. Limited, 978-81-265-1771-8, BOOK, Pierre BrÃ©maud, Fourier Analysis and Stochastic Processes,weblink 16 September 2014, Springer, 978-3-319-09590-5, BOOK, Adam Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction,weblink 11 August 2005, Cambridge University Press, 978-0-521-83166-6, The theory of stochastic processes is considered to be an important contribution to mathematicsJOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1336â€“1347, and it continues to be an active topic of research for both theoretical reasons and applications.BOOK, Jochen Blath, Peter Imkeller, Sylvie RÅ“lly, Surveys in Stochastic Processes,weblink 2011, European Mathematical Society, 978-3-03719-072-2, BOOK, Michel Talagrand, Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems,weblink 12 February 2014, Springer Science & Business Media, 978-3-642-54075-2, 4â€“, BOOK, Paul C. Bressloff, Stochastic Processes in Cell Biology,weblink 22 August 2014, Springer, 978-3-319-08488-6, viiâ€“ix,

## 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.)

### Classifications

A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 26, 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, 24 and 25, When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in**discrete time**.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 527, If the index set is some interval of the real line, then time is said to be

**continuous**. The two types of stochastic processes are respectively referred to as

**discrete-time**and

**continuous-time stochastic processes**.BOOK, Jeffrey S Rosenthal, A First Look at Rigorous Probability Theory,weblink 14 November 2006, World Scientific Publishing Co Inc, 978-981-310-165-4, 177â€“178, Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.BOOK, Peter E. Kloeden, Eckhard Platen, Numerical Solution of Stochastic Differential Equations,weblink 17 April 2013, Springer Science & Business Media, 978-3-662-12616-5, 63, BOOK, Davar Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields,weblink 10 April 2006, Springer Science & Business Media, 978-0-387-21631-7, 153â€“155, If the index set is the integers, or some subset of them, then the stochastic process can also be called a

**random sequence**.If the state space is the integers or natural numbers, then the stochastic process is called a

**discrete**or

**integer-valued stochastic process**. If the state space is the real line, then the stochastic process is referred to as a

**real-valued stochastic process**or a

**process with continuous state space**. If the state space is n-dimensional Euclidean space, then the stochastic process is called a n-

**dimensional vector process**or n-

**vector process**.

### 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,

### Terminology

The definition of a stochastic process varies,BOOK, Bert E. Fristedt, Lawrence F. Gray, A Modern Approach to Probability Theory,weblink 21 November 2013, Springer Science & Business Media, 978-1-4899-2837-5, 580, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The terms*random process*and

*stochastic process*are considered synonyms and are used interchangeably, without the index set being precisely specified.BOOK, David Stirzaker, Stochastic Processes and Models,weblink 2005, Oxford University Press, 978-0-19-856814-8, 45, BOOK, Murray Rosenblatt, Random Processes,weblink 1962, Oxford University Press, 91, BOOK, John A. Gubner, Probability and Random Processes for Electrical and Computer Engineers,weblink 1 June 2006, Cambridge University Press, 978-1-139-45717-0, 383, Both "collection", or "family" are usedBOOK, Kiyosi ItÅ, Essentials of Stochastic Processes,weblink 2006, American Mathematical Soc., 978-0-8218-3898-3, 13, while instead of "index set", sometimes the terms "parameter set" or "parameter space" are used.The term

*random function*is also used to refer to a stochastic or random process,BOOK, M. LoÃ¨ve, Probability Theory II,weblink 15 May 1978, Springer Science & Business Media, 978-0-387-90262-3, 163, BOOK, Pierre BrÃ©maud, Fourier Analysis and Stochastic Processes,weblink 16 September 2014, Springer, 978-3-319-09590-5, 133, though sometimes it is only used when the stochastic process takes real values. This term is also used when the index sets are mathematical spaces other than the real line,BOOK, Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory,weblink 10 July 2010, Springer Science & Business Media, 978-0-387-87862-1, 1, while the terms

*stochastic process*and

*random process*are usually used when the index set interpreted as time,BOOK, Richard F. Bass, Stochastic Processes,weblink 6 October 2011, Cambridge University Press, 978-1-139-50147-7, 1, and other terms are used such as

*random field*when the index set is n-dimensional Euclidean space R^n or a manifold.

### 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,### Random walk

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.BOOK, Achim Klenke, Probability Theory: A Comprehensive Course,weblink 30 September 2013, Springer, 978-1-4471-5362-7, 347, BOOK, Gregory F. Lawler, Vlada Limic, Random Walk: A Modern Introduction,weblink 24 June 2010, Cambridge University Press, 978-1-139-48876-1, 1, BOOK, Olav Kallenberg, Foundations of Modern Probability,weblink 8 January 2002, Springer Science & Business Media, 978-0-387-95313-7, 136, BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 383, BOOK, Rick Durrett, Probability: Theory and Examples,weblink 30 August 2010, Cambridge University Press, 978-1-139-49113-6, 277, But some also use the term to refer to processes that change in continuous time,BOOK, Weiss, George H., Encyclopedia of Statistical Sciences, Random Walks, 2006, 10.1002/0471667196.ess2180.pub2, 1, 978-0471667193, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.BOOK, Aris Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Observational Data,weblink 2 September 1999, Cambridge University Press, 978-0-521-42408-0, 454, There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 81, A classic example of a random walk is known as the*simple random walk*, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p, or decreases by one with probability 1-p, so index set of this random walk is the natural numbers, while its state space is the integers. If the p=0.5, this random walk is called a symmetric random walk.BOOK, Allan Gut, Probability: A Graduate Course,weblink 17 October 2012, Springer Science & Business Media, 978-1-4614-4708-5, 88, BOOK, Geoffrey Grimmett, David Stirzaker, Probability and Random Processes,weblink 31 May 2001, OUP Oxford, 978-0-19-857222-0, 71,

### Wiener process

The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments.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, 1, BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 56, The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.JOURNAL, Brush, Stephen G., A history of random processes, Archive for History of Exact Sciences, 5, 1, 1968, 1â€“2, 0003-9519, 10.1007/BF00328110, JOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1338, BOOK, Iosif Ilyich Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, 21, (File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).)Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 29, BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 471, BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 21 and 22, BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, VIII, BOOK, Daniel Revuz, Marc Yor, Continuous Martingales and Brownian Motion,weblink 9 March 2013, Springer Science & Business Media, 978-3-662-06400-9, IX, Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.BOOK, Jeffrey S Rosenthal, A First Look at Rigorous Probability Theory,weblink 14 November 2006, World Scientific Publishing Co Inc, 978-981-310-165-4, 186, But the process can be defined more generally so its state space can be n-dimensional Euclidean space.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, 33, If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant mu, which is a real number, then the resulting stochastic process is said to have drift mu.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 118, BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, 78, Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk.JOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1337, BOOK, Peter MÃ¶rters, Yuval Peres, Brownian Motion,weblink 25 March 2010, Cambridge University Press, 978-1-139-48657-6, 1 and 3, The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, 61, BOOK, Steven E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models,weblink 3 June 2004, Springer Science & Business Media, 978-0-387-40101-0, 93, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.BOOK, Olav Kallenberg, Foundations of Modern Probability,weblink 8 January 2002, Springer Science & Business Media, 978-0-387-95313-7, 225 and 260, BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, 70, BOOK, Peter MÃ¶rters, Yuval Peres, Brownian Motion,weblink 25 March 2010, Cambridge University Press, 978-1-139-48657-6, 131, The Wiener process is a member of some important families of stochastic processes, including Markov processes, LÃ©vy processes and Gaussian processes. The process also has many applications and is the main stochastic process used in stochastic calculus.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, It plays a central role in quantitative finance,JOURNAL, Applebaum, David, LÃ©vy processes: From probability to finance and quantum groups, Notices of the AMS, 51, 11, 2004, 1341, BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 340, where it is used, for example, in the Blackâ€“Scholesâ€“Merton model.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 124, The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, 47, BOOK, Ubbo F. Wiersema, Brownian Motion Calculus,weblink 6 August 2008, John Wiley & Sons, 978-0-470-02171-2, 2,### Poisson process

The Poisson process is a stochastic process that has different forms and definitions.BOOK, Henk C. Tijms, A First Course in Stochastic Models,weblink 6 May 2003, Wiley, 978-0-471-49881-0, 1 and 2, 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, 19â€“36, It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.BOOK, Mark A. Pinsky, Samuel Karlin, An Introduction to Stochastic Modeling,weblink 2011, Academic Press, 978-0-12-381416-6, 241, The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and LÃ©vy processes.The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.BOOK, J. F. C. Kingman, Poisson Processes,weblink 17 December 1992, Clarendon Press, 978-0-19-159124-2, 38, 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, 19, If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.BOOK, J. F. C. Kingman, Poisson Processes,weblink 17 December 1992, Clarendon Press, 978-0-19-159124-2, 22, Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 118 and 119, BOOK, Leonard Kleinrock, Queueing Systems: Theory,weblink 1976, Wiley, 978-0-471-49110-1, 61, Defined on the real line, the Poisson process can be interpreted as a stochastic process,BOOK, Murray Rosenblatt, Random Processes,weblink 1962, Oxford University Press, 94, among other random objects.BOOK, Martin Haenggi, Stochastic Geometry for Wireless Networks,weblink 2013, Cambridge University Press, 978-1-107-01469-5, 10 and 18, BOOK, Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke, Stochastic Geometry and Its Applications,weblink 27 June 2013, John Wiley & Sons, 978-1-118-65825-3, 41 and 108, But then it can be defined on the n-dimensional Euclidean space or other mathematical spaces,BOOK, J. F. C. Kingman, Poisson Processes,weblink 17 December 1992, Clarendon Press, 978-0-19-159124-2, 11, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.BOOK, Roy L. Streit, Poisson Point Processes: Imaging, Tracking, and Sensing,weblink 15 September 2010, Springer Science & Business Media, 978-1-4419-6923-1, 1, But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.BOOK, J. F. C. Kingman, Poisson Processes,weblink 17 December 1992, Clarendon Press, 978-0-19-159124-2, v,### Further examples

#### Markov processes and chains

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.BOOK, Richard Serfozo, Basics of Applied Stochastic Processes,weblink 24 January 2009, Springer Science & Business Media, 978-3-540-89332-5, 2, BOOK, Y.A. Rozanov, Markov Random Fields,weblink 6 December 2012, Springer Science & Business Media, 978-1-4613-8190-7, 58, The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processesBOOK, Sheldon M. Ross, Stochastic processes,weblink 1996, Wiley, 978-0-471-12062-9, 235 and 358, in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 373 and 374, BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 49, A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies.BOOK,weblink Applied Probability and Queues, 15 May 2003, Springer Science & Business Media, 978-0-387-00211-8, 7, SÃ¸ren Asmussen, For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time),BOOK,weblink Stochastic Processes, 17 June 2015, Courier Dover Publications, 978-0-486-79688-8, 188, Emanuel Parzen, BOOK,weblink A First Course in Stochastic Processes, 2 December 2012, Academic Press, 978-0-08-057041-9, 29 and 30, Samuel Karlin, Howard E. Taylor, BOOK,weblink Stochastic processes: a survey of the mathematical theory, Springer-Verlag, 1977, 978-3-540-90275-1, 106â€“121, John Lamperti, BOOK,weblink Stochastic processes, Wiley, 1996, 978-0-471-12062-9, 174 and 231, Sheldon M. Ross, but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used used by researchers like Joseph Doob and Kai Lai Chung.BOOK, Sean Meyn, Richard L. Tweedie, Markov Chains and Stochastic Stability,weblink 2 April 2009, Cambridge University Press, 978-0-521-73182-9, 19, Markov processes form an important class of stochastic processes and have applications in many areas.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 47, For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.BOOK, Reuven Y. Rubinstein, Dirk P. Kroese, Simulation and the Monte Carlo Method,weblink 20 September 2011, John Wiley & Sons, 978-1-118-21052-9, 225, BOOK, Dani Gamerman, Hedibert F. Lopes, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition,weblink 10 May 2006, CRC Press, 978-1-58488-587-0, The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as n-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.BOOK, Y.A. Rozanov, Markov Random Fields,weblink 6 December 2012, Springer Science & Business Media, 978-1-4613-8190-7, 61, 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, 27, BOOK, Pierre Bremaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,weblink 9 March 2013, Springer Science & Business Media, 978-1-4757-3124-8, 253,#### Martingale

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 65, BOOK, Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus,weblink 1991, Springer, 978-1-4612-0949-2, 11, BOOK, David Williams, Probability with Martingales,weblink 14 February 1991, Cambridge University Press, 978-0-521-40605-5, 93 and 94, but they can also be complex-valuedBOOK, Joseph L. Doob, Stochastic processes,weblink 1990, Wiley, 292 and 293, or even more general.BOOK, Gilles Pisier, Martingales in Banach Spaces,weblink 6 June 2016, Cambridge University Press, 978-1-316-67946-3, A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a sequence of independent and identically distributed random variables X_1, X_2, X_3, dots with zero mean, the stochastic process formed from the successive partial sums X_1,X_1+ X_2, X_1+ X_2+X_3, dots is a discrete-time martingale.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 12 and 13, In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.BOOK, P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application,weblink 10 July 2014, Elsevier Science, 978-1-4832-6322-9, 2, Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the*compensated Poisson process*. Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 115, Martingales mathematically formalize the idea of a fair game,BOOK, Sheldon M. Ross, Stochastic processes,weblink 1996, Wiley, 978-0-471-12062-9, 295, and they were originally developed to show that it is not possible to win a fair game. But now they are used in many areas of probability, which is one of the main reasons for studying them.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 11, BOOK, Olav Kallenberg, Foundations of Modern Probability,weblink 8 January 2002, Springer Science & Business Media, 978-0-387-95313-7, 96, Many problems in probability have been solved by finding a martingale in the problem and studying it.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 371, Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 22, BOOK, Geoffrey Grimmett, David Stirzaker, Probability and Random Processes,weblink 31 May 2001, OUP Oxford, 978-0-19-857222-0, 336, Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.JOURNAL, Glasserman, Paul, Kou, Steven, A Conversation with Chris Heyde, Statistical Science, 21, 2, 2006, 292 and 293, 0883-4237, 10.1214/088342306000000088, math/0609294, They have found applications in areas in probability theory such as queueing theory and Palm calculusBOOK, Francois Baccelli, Pierre Bremaud, Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences,weblink 11 November 2013, Springer Science & Business Media, 978-3-662-11657-9, and other fields such as economicsBOOK, P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application,weblink 10 July 2014, Elsevier Science, 978-1-4832-6322-9, x, and finance.

#### LÃ©vy process

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,#### Point process

A point process is a collection of points randomly located on some mathematical space such as the real line, n-dimensional Euclidean space, or more abstract spaces. Sometimes the term*point process*is not preferred, as historically the word

*process*denoted an evolution of some system in time, so a point process is also called a

**random point field**.BOOK, Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke, Stochastic Geometry and Its Applications,weblink 27 June 2013, John Wiley & Sons, 978-1-118-65825-3, 109, There are different interpretations of a point process, such a random counting measure or a random set.BOOK, Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke, Stochastic Geometry and Its Applications,weblink 27 June 2013, John Wiley & Sons, 978-1-118-65825-3, 108, BOOK, Martin Haenggi, Stochastic Geometry for Wireless Networks,weblink 2013, Cambridge University Press, 978-1-107-01469-5, 10, Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic 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, 194, BOOK, D.R. Cox, Valerie Isham, Point Processes,weblink 17 July 1980, CRC Press, 978-0-412-21910-8, 3, though it has been remarked that the difference between point processes and stochastic processes is not clear.Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space{{efn|In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line,BOOK, J. F. C. Kingman, Poisson Processes,weblink 17 December 1992, Clarendon Press, 978-0-19-159124-2, 8, BOOK, Jesper Moller, Rasmus Plenge Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes,weblink 25 September 2003, CRC Press, 978-0-203-49693-0, 7, which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or n-dimensional Euclidean space.BOOK, Samuel Karlin, Howard E. Taylor, A First Course in Stochastic Processes,weblink 2 December 2012, Academic Press, 978-0-08-057041-9, 31, BOOK, Volker Schmidt, Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms,weblink 24 October 2014, Springer, 978-3-319-10064-7, 99, Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.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, BOOK, D.R. Cox, Valerie Isham, Point Processes,weblink 17 July 1980, CRC Press, 978-0-412-21910-8,

## 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 set**BOOK, 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 function**BOOK, 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.

#### Stationarity

**Stationarity**is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if X is a stationary stochastic process, then for any tin T the random variable X_t has the same distribution, which means that for any set of n index set values t_1,dots, t_n, the corresponding n random variablesX_{t_1}, dots X_{t_n},all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.BOOK, John Lamperti, Stochastic processes: a survey of the mathematical theory,weblink 1977, Springer-Verlag, 978-3-540-90275-1, 6 and 7, BOOK, Iosif I. Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, 4, But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.BOOK, Robert J. Adler, The Geometry of Random Fields,weblink 28 January 2010, SIAM, 978-0-89871-693-1, 14 and 15, BOOK, Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke, Stochastic Geometry and Its Applications,weblink 27 June 2013, John Wiley & Sons, 978-1-118-65825-3, 112, When the index set T can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.BOOK, Joseph L. Doob, Stochastic processes,weblink 1990, Wiley, 94â€“96, A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process X is said to be stationary in the wide sense, then the process X has a finite second moment for all tin T and the covariance of the two random variables X_t and X_{t+h} depends only on the number h for all tin T.BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 298 and 299, Khinchin introduced the related concept of

**stationarity in the wide sense**, which has other names including

**covariance stationarity**or

**stationarity in the broad sense**.BOOK, Iosif Ilyich Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, 8,

#### 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,#### Modification

A**modification**of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process X that has the same index set T, set space S, and probability space (Omega,{cal F},P) as another stochastic process Y is said to be a modification of Y if for all tin T the followingP(X_t=Y_t)=1 ,holds. Two stochastic processes that are modifications of each other have the same lawBOOK, 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, 130, and they are said to be

**stochastically equivalent**or

**equivalent**.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 530, Instead of modification, the term

**version**is also used,BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 48, BOOK, Bernt Ã˜ksendal, Stochastic Differential Equations: An Introduction with Applications,weblink 2003, Springer Science & Business Media, 978-3-540-04758-2, 14, BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 472, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.BOOK, Daniel Revuz, Marc Yor, Continuous Martingales and Brownian Motion,weblink 9 March 2013, Springer Science & Business Media, 978-3-662-06400-9, 18â€“19, If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.BOOK, David Applebaum, LÃ©vy Processes and Stochastic Calculus,weblink 5 July 2004, Cambridge University Press, 978-0-521-83263-2, 20, The theorem can also be generalized to random fields so the index set is n-dimensional Euclidean spaceBOOK, Hiroshi Kunita, Stochastic Flows and Stochastic Differential Equations,weblink 3 April 1997, Cambridge University Press, 978-0-521-59925-2, 31, as well as to stochastic processes with metric spaces as their state spaces.BOOK, Olav Kallenberg, Foundations of Modern Probability,weblink 8 January 2002, Springer Science & Business Media, 978-0-387-95313-7, 35,

#### 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,

#### Separability

**Separability**is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space,{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.}} which means that the index set has a dense countable subset.BOOK, Kiyosi ItÅ, Essentials of Stochastic Processes,weblink 2006, American Mathematical Soc., 978-0-8218-3898-3, 32â€“33, More precisely, a real-valued continuous-time stochastic process X with a probability space (Omega,{cal F},P) is separable if its index set T has a dense countable subset Usubset T and there is a set Omega_0 subset Omega of probability zero, so P(Omega_0)=0, such that for every open set Gsubset T and every closed set Fsubset textstyle R =(-infty,infty) , the two events { X_t in F text{ for all } t in Gcap U} and { X_t in F text{ for all } t in G} differ from each other at most on a subset of Omega_0.BOOK, Iosif Ilyich Gikhman, Anatoly Vladimirovich Skorokhod, Introduction to the Theory of Random Processes,weblink 1969, Courier Corporation, 978-0-486-69387-3, 150, BOOK, Petar Todorovic, An Introduction to Stochastic Processes and Their Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4613-9742-7, 19â€“20, BOOK, Ilya Molchanov, Theory of Random Sets,weblink 11 May 2005, Springer Science & Business Media, 978-1-85233-892-3, 340, The definition of separability{{efn|The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.BOOK, Patrick Billingsley, Probability and Measure,weblink 4 August 2008, Wiley India Pvt. Limited, 978-81-265-1771-8, 526â€“527, BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 535, }} can also be stated for other index sets and state spaces,BOOK, Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory,weblink 10 July 2010, Springer Science & Business Media, 978-0-387-87862-1, 22, such as in the case of random fields, where the index set as well as the state space can be n-dimensional Euclidean space.The concept of separability of a stochastic process was introduced by Joseph Doob, where the underlying idea is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.BOOK, Joseph L. Doob, Stochastic processes,weblink 1990, Wiley, 56, A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.BOOK, Davar Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields,weblink 10 April 2006, Springer Science & Business Media, 978-0-387-21631-7, 155, Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

#### 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:

left{X_tright},left{Y_tright} text{ uncorrelated} quad iff quad operatorname{K}_{mathbf{X}mathbf{Y}}(t_1,t_2) = 0 quad forall t_1,t_2.

#### 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:

left{X_tright},left{Y_tright} text{ orthogonal} quad iff quad operatorname{R}_{mathbf{X}mathbf{Y}}(t_1,t_2) = 0 quad forall t_1,t_2.

#### Skorokhod space

A**Skorokhod space**, also written as

**Skorohod space**, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as [0,1] or [0,infty), and take values on the real line or on some metric space.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, 78â€“79, BOOK, Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory,weblink 10 July 2010, Springer Science & Business Media, 978-0-387-87862-1, 24, BOOK, Vladimir I. Bogachev, Measure Theory (Volume 2),weblink 15 January 2007, Springer Science & Business Media, 978-3-540-34514-5, 53, Such functions are known as cÃ dlÃ g or cadlag functions, based on the acronym of the French expression

*continue Ã droite, limite Ã gauche*, due to the functions being right-continuous with left limits.BOOK, Fima C. Klebaner, Introduction to Stochastic Calculus with Applications,weblink 2005, Imperial College Press, 978-1-86094-555-7, 4, A Skorokhod function space, introduced by Anatoliy Skorokhod, is often denoted with the letter D, so the function space is also referred to as space D.BOOK, SÃ¸ren Asmussen, Applied Probability and Queues,weblink 15 May 2003, Springer Science & Business Media, 978-0-387-00211-8, 420, BOOK, Patrick Billingsley, Convergence of Probability Measures,weblink 25 June 2013, John Wiley & Sons, 978-1-118-62596-5, 121, The notation of this function space can also include the interval on which all the cÃ dlÃ g functions are defined, so, for example, D[0,1] denotes the space of cÃ dlÃ g functions defined on the unit interval [0,1].BOOK, Richard F. Bass, Stochastic Processes,weblink 6 October 2011, Cambridge University Press, 978-1-139-50147-7, 34, Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.BOOK, Nicholas H. Bingham, RÃ¼diger Kiesel, Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives,weblink 29 June 2013, Springer Science & Business Media, 978-1-4471-3856-3, 154,

#### 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. )

### Stochastic processes after World War II

After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.JOURNAL, Meyer, Paul-AndrÃ©, Stochastic Processes from 1950 to the Present, Electronic Journal for History of Probability and Statistics, 5, 1, 2009, 1â€“42, Starting in the 1940s, Kiyosi ItÃ´ published papers developing the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on the Wiener or Brownian motion process.JOURNAL, Kiyosi ItÃ´ receives Kyoto Prize, Notices of the AMS, 45, 8, 1998, 981â€“982, Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob. Further work, considered pioneering, was done by Gilbert Hunt in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of LÃ©vy processes and led to more interest in studying Markov processes with methods developed by ItÃ´.BOOK, Jean Bertoin, LÃ©vy Processes,weblink 29 October 1998, Cambridge University Press, 978-0-521-64632-1, viii and ix, BOOK, J. Michael Steele, Stochastic Calculus and Financial Applications,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-9305-4, 176, In 1953 Doob published his book*Stochastic processes*, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.JOURNAL, Bingham, N. H., Doob: a half-century on, Journal of Applied Probability, 42, 1, 2005, 257â€“266, 0021-9002, 10.1239/jap/1110381385, Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-AndrÃ© Meyer. Earlier work had been carried out by Sergei Bernstein, Paul LÃ©vy and Jean Ville, the latter adopting the term martingale for the stochastic process.BOOK, P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application,weblink 10 July 2014, Elsevier Science, 978-1-4832-6322-9, 1 and 2, JOURNAL, Dynkin, E. B., Kolmogorov and the Theory of Markov Processes, The Annals of Probability, 17, 3, 1989, 822â€“832, 0091-1798, 10.1214/aop/1176991248, Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations. The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and Monroe D. Donsker and Srinivasa Varadhan in the United States of America,JOURNAL, Ellis, Richard S., An overview of the theory of large deviations and applications to statistical mechanics, Scandinavian Actuarial Journal, 1995, 1, 1995, 98, 0346-1238, 10.1080/03461238.1995.10413952, which would later result in Varadhan winning the 2007 Abel Prize.JOURNAL, Raussen, Martin, Skau, Christian, Interview with Srinivasa Varadhan, Notices of the AMS, 55, 2, 2008, 238â€“246, In the 1990s and 2000s the theories of Schrammâ€“Loewner evolutionBOOK, Malte Henkel, Dragi Karevski, Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution,weblink 4 April 2012, Springer Science & Business Media, 978-3-642-27933-1, 113, and rough pathsBOOK, Peter K. Friz, Nicolas B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications,weblink 4 February 2010, Cambridge University Press, 978-1-139-48721-4, 571, were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin WernerJOURNAL, 2006 Fields Medals Awarded, Notices of the AMS, 53, 9, 2015, 1041â€“1044, in 2008 and to Martin Hairer in 2014.JOURNAL, Quastel, Jeremy, The Work of the 2014 Fields Medalists, Notices of the AMS, 62, 11, 2015, 1341â€“1344, The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.

### 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,

#### Random walks

In 1905 Karl Pearson coined the term*random walk*while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. For example, the problem known as the

*Gambler's ruin*is based on a simple random walk,BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 374, and is an example of a random walk with absorbing barriers.BOOK, Oliver C. Ibe, Elements of Random Walk and Diffusion Processes,weblink 29 August 2013, John Wiley & Sons, 978-1-118-61793-9, 5, Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods,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, 63, and then more detailed solutions were presented by Jakob Bernoulli and Abraham de Moivre.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, 202, For random walks in n-dimensional integer lattices, George PÃ³lya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. PÃ³lya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.BOOK, Ionut Florescu, Probability and Stochastic Processes,weblink 7 November 2014, John Wiley & Sons, 978-1-118-59320-2, 385, BOOK, Barry D. Hughes, Random Walks and Random Environments: Random walks,weblink 1995, Clarendon Press, 978-0-19-853788-5, 111,

#### 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

Markov processes and Markov chains are named after Andrey Markov who studied Markov chains in the early 20th century.BOOK, Markov Chains: From Theory to Implementation and Experimentation, Gagniuc, Paul A., John Wiley & Sons, 2017, 978-1-119-38755-8, USA, NJ, 2â€“8, Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption,BOOK, Charles Miller Grinstead, James Laurie Snell, Introduction to Probability,weblink 1997, American Mathematical Soc., 978-0-8218-0749-1, 464â€“466, BOOK, Pierre Bremaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,weblink 9 March 2013, Springer Science & Business Media, 978-1-4757-3124-8, ix, JOURNAL, Hayes, Brian, First links in the Markov chain, American Scientist, 101, 2, 2013, 92â€“96, 10.1511/2013.101.92, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains.In 1912 PoincarÃ© studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by IrÃ©nÃ©e-Jules BienaymÃ©.JOURNAL, Seneta, E., I.J. BienaymÃ© [1796-1878]: Criticality, Inequality, and Internationalization, International Statistical Review / Revue Internationale de Statistique, 66, 3, 1998, 291â€“292, 0306-7734, 10.2307/1403518, 1403518, Starting in 1928, Maurice FrÃ©chet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.BOOK, Bru, B., Statisticians of the Centuries, Hertz, S., Maurice FrÃ©chet, 2001, 331â€“334, 10.1007/978-1-4613-0179-0_71, 978-0-387-95283-3, Andrei Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement.BOOK, Marc Barbut, Bernard Locker, Laurent Mazliak, Paul LÃ©vy and Maurice FrÃ©chet: 50 Years of Correspondence in 107 Letters,weblink 23 August 2016, Springer London, 978-1-4471-7262-8, 5, He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.BOOK, Valeriy Skorokhod, Basic Principles and Applications of Probability Theory,weblink 5 December 2005, Springer Science & Business Media, 978-3-540-26312-8, 146, Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapmanâ€“Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.JOURNAL, Bernstein, Jeremy, Bachelier, American Journal of Physics, 73, 5, 2005, 398â€“396, 0002-9505, 10.1119/1.1848117, 2005AmJPh..73..395B, The differential equations are now called the Kolmogorov equationsBOOK, William J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-3038-0, vii, or the Kolmogorovâ€“Chapman equations.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, 57, 0024-6093, 10.1112/blms/22.1.31, Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.#### LÃ©vy 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:- a sample function of a stochastic process X is a continuous function of tin T;
- a sample function of a stochastic process X is a bounded function of tin T; and
- a sample function of a stochastic process X is an increasing function of tin T.

### Resolving construction issues

One approach for avoiding mathematical construction issues of stochastic processes, proposed by Joseph Doob, is to assume that the stochastic process is separable.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, 221, Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set.BOOK, Robert J. Adler, Jonathan E. Taylor, Random Fields and Geometry,weblink 29 January 2009, Springer Science & Business Media, 978-0-387-48116-6, 14, Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov,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, 211, for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption,JOURNAL, Getoor, Ronald, J. L. Doob: Foundations of stochastic processes and probabilistic potential theory, The Annals of Probability, 37, 5, 2009, 1655, 0091-1798, 10.1214/09-AOP465, 0909.4213, but such a stochastic process based on this approach will be automatically separable.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 536, Although less used, the separability assumption is considered more general because every stochastic process has a separable version. It is also used when it is not possible to construct a stochastic process in a Skorokhod space.BOOK, Alexander A. Borovkov, Probability Theory,weblink 22 June 2013, Springer Science & Business Media, 978-1-4471-5201-9, 535, For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as n-dimensional Euclidean space.BOOK, Benjamin Yakir, Extremes in Random Fields: A Theory and Its Applications,weblink 1 August 2013, John Wiley & Sons, 978-1-118-72062-2, 5,## See also

{hide}columns-list|colwidth=30em|- List of stochastic processes topics
- Covariance function
- Deterministic system
- Dynamics of Markovian particles
- Entropy rate (for a stochastic process)
- Ergodic process
- GenI process
- Gillespie algorithm
- Interacting particle system
- Law (stochastic processes)
- Markov chain
- Probabilistic cellular automaton
- Random field
- Randomness
- Stationary process
- Statistical model
- Stochastic calculus
- Stochastic control

## Notes

{{Notelist}}## References

{{Reflist}}## Further reading

{{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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