Stochastic process
In probability theory and related fields, a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.[1][4][5] Stochastic processes have applications in many disciplines such as biology,[6] chemistry,[7] ecology,[8] neuroscience,[9] physics,[10] image processing, signal processing,[11] control theory,[12] information theory,[13] computer science,[14] and telecommunications.[15] Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.[16][17][18]
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,[a] used by Louis Bachelier to study price changes on the Paris Bourse,[21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.[22] These two stochastic processes are considered the most important and central in the theory of stochastic processes,[1][4][23] and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.[21][24]
The term random function is also used to refer to a stochastic or random process,[25][26] because a stochastic process can also be interpreted as a random element in a function space.[27][28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables.[27][29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line.[5][29] 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.[5][30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.[5][28]
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks,[31] martingales,[32] Markov processes,[33] Lévy processes,[34] Gaussian processes,[35] random fields,[36] renewal processes, and branching processes.[37] The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology[38][39][40] as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.[41][42][43] The theory of stochastic processes is considered to be an important contribution to mathematics[44] and it continues to be an active topic of research for both theoretical reasons and applications.[45][46][47]
Definitions[edit]
Stochastic process[edit]
A stochastic process is defined as a collection of random variables defined on a common probability space , where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .[28]
In other words, for a given probability space and a measurable space , a stochastic process is a collection of -valued random variables, which can be written as:[80]
History[edit]
Early probability theory[edit]
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,[239] but very little analysis on them was done in terms of probability.[240] 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.[241][242] 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.[243]
After Cardano, Jakob Bernoulli[e] 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.[245][246] But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev,[247][248] most of the mathematical community[f] did not consider probability theory to be part of mathematics until the 20th century.[247][249][250][251]
Statistical mechanics[edit]
In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, are 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.[252][253] 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 modelled the gas particles as moving in random directions at random velocities.[254][255] 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.[256]
Measure theory and probability theory[edit]
At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.[248] 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 Émile Borel. In 1925, another French mathematician Paul Lévy published the first probability book that used ideas from measure theory.[248]
In the 1920s, fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein, Aleksandr Khinchin,[g] and Andrei Kolmogorov.[251] Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.[257] In the early 1930s, Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov,[258] and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.[63][259][h]
Birth of modern probability theory[edit]
In 1933, Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,[i] 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.[248][251]
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 Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.[248][251]
Decades later, Cramér referred to the 1930s as the "heroic period of mathematical probability theory".[251] 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[251] and the death of Doeblin, considered now a pioneer in stochastic processes.[261]
a sample function of a stochastic process is a of ;
continuous function
a sample function of a stochastic process is a of ; and
bounded function
a sample function of a stochastic process is an of .
