Stochastic process

Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

[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.

[28][50] 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.

[61] This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz[62] who in 1917 wrote in German the word stochastik with a sense meaning random.

[65] According to the Oxford English Dictionary, 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.)".

[90] There are various other 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.

[22][131] 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.

[168][170][176] Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

[211] A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.

[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.

[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.

[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] 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.

[265] 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.

Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process.

[21] In 1880, Danish astronomer 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.

[286] The French mathematician Louis Bachelier used a Wiener process in his 1900 thesis[287][288] in order to model price changes on the Paris Bourse, a stock exchange,[289] without knowing the work of Thiele.

But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas,[290] which was cited by mathematicians including Doob, Feller[290] and Kolmogorov.

[22][24] 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.

Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval.

[296][297] 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 Irénée-Jules Bienaymé.

This result was later derived under more general conditions by Lévy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937.

[307][313] 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.

Developed by Fischer Black, Myron Scholes, and Robert Solow, this model uses Geometric Brownian motion, a specific type of stochastic process, to describe the dynamics of asset prices.

The birth-death process,[322] a simple stochastic model, describes how populations fluctuate over time due to random births and deaths.

For instance, Markov chains are widely used in probabilistic algorithms for optimization and sampling tasks, such as those employed in search engines like Google's PageRank.

Randomized algorithms are also extensively applied in areas such as cryptography, large-scale simulations, and artificial intelligence, where uncertainty must be managed effectively.

[323] Another significant application of stochastic processes in computer science is in queuing theory, which models the random arrival and service of tasks in a system.

For instance, queuing models help predict delays, manage resource allocation, and optimize throughput in web servers and communication networks.