Stochasticity is used in many different fields, including the natural sciences such as biology,[7] chemistry,[8] ecology,[9] neuroscience,[10] and physics,[11] as well as technology and engineering fields such as image processing, signal processing,[12] information theory,[13] computer science,[14] cryptography,[15] and telecommunications.[16] It is also used in finance, due to seemingly random changes in financial markets[17][18][19] as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.
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.[1] 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".[20] This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz,[21] 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.[1] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin,[22][23] though the German term had been used earlier in 1931 by Andrey Kolmogorov.[24]
The Monte Carlo method is a stochastic method popularized by physics researchers StanisÅaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis.[33] The use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino.Methods of simulation and statistical sampling generally did the opposite: using simulation to test a previously understood deterministic problem. Though examples of an "inverted" approach do exist historically, they were not considered a general method until the popularity of the Monte Carlo method spread.
Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method to calculate the properties of the newly discovered neutron. Monte Carlo methods were central to the simulations required for the Manhattan Project, though they were severely limited by the computational tools of the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The RAND Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.
The marketing and the changing movement of audience tastes and preferences, as well as the solicitation of and the scientific appeal of certain film and television debuts (i.e., their opening weekends, word-of-mouth, top-of-mind knowledge among surveyed groups, star name recognition and other elements of social media outreach and advertising), are determined in part by stochastic modeling. A recent attempt at repeat business analysis was done by Japanese scholars[citation needed] and is part of the Cinematic Contagion Systems patented by Geneva Media Holdings, and such modeling has been used in data collection from the time of the original Nielsen ratings to modern studio and television test audiences.
Stochastic processes may be used in music to compose a fixed piece or may be produced in performance. Stochastic music was pioneered by Iannis Xenakis, who coined the term stochastic music. Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST/10 and AtrÃes, Markov chains in Analogiques, game theory in Duel and StratÃgie, group theory in Nomos Alpha (for Siegfried Palm), set theory in Herma and Eonta,[41] and Brownian motion in N'Shima.[citation needed] Xenakis frequently used computers to produce his scores, such as the ST series including Morsima-Amorsima and AtrÃes, and founded CEMAMu. Earlier, John Cage and others had composed aleatoric or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching). Lejaren Hiller and Leonard Issacson used generative grammars and Markov chains in their 1957 Illiac Suite. Modern electronic music production techniques make these processes relatively simple to implement, and many hardware devices such as synthesizers and drum machines incorporate randomization features. Generative music techniques are therefore readily accessible to composers, performers, and producers.
Statistical Inference for Stochastic Processes is an international journal publishing articles on parametric and nonparametric inference for discrete- and continuous-time stochastic processes, and their applications to biology, chemistry, physics, finance, economics, and other sciences.
The probability research group is primarily focused on discrete probability topics. Random graphs and percolation models (infinite random graphs) are studied using stochastic ordering, subadditivity, and the probabilistic method, and have applications to phase transitions and critical phenomena in physics, flow of fluids in porous media, and spread of epidemics or knowledge in populations. Convergence rates to equilibrium in Markov chains are studied and applied to Markov Chain Monte Carlo simulation, and related algorithms for perfect sampling are created and analyzed.Various probabilistic and other techniques are used to analyze the performance of algorithms in computer science used for such purposes as sorting and searching. A plethora of interesting questions and applications allow us to involve both undergraduate and graduate students in valuable research in modern probability and stochastic processes.
Stochastic processes are probabilistic models for random quantities evolving in time or space. The evolution is governed by some dependence relationship between the random quantities at different times or locations. Major classes of stochastic processes are random walks, Markov processes, branching processes, renewal processes, martingales, and Brownian motion. Important application areas are mathematical finance, queuing processes, analysis of computer algorithms, economic time series, image analysis, social networks, and modeling biomedical phenomena. Stochastic process models are used extensively in operations research applications.
The current work, led by Kiyoshi Kanazawa and overseen by Takayasu and others, benefited from a more comprehensive dataset that became available in July 2016. This dataset enabled a meticulous approach to tracking the trend-following behavior[3] of individual traders. When examined collectively, this trend-following was found to be analogous to the concept of inertia[4] in physics.
It is not a rigorous book, but it requires no previous experience with stochastic processes. Actually I would consider it a good introduction to stochastic processes. I found all the physics presented prior to the finance section to be interesting and relevant.
Physicists are motivated to study financial markets in the expectation that stochastic analysis developed in the context of statistical mechanics and nonlinear physics may find applications there. The relevant issues are the characteristics of the stochastic processes underlying financial markets. Recent analyses of financial markets have shown that the process is highly complex and that the intensity of fluctuations depends on the time of day and on variations in price during the day. Our research involves modeling and stochastic analyses of such processes.
Stochastic calculus provides a powerful description of a specific class of stochastic processes in physics and finance. However, many econophysicists struggle to understand it. This book presents the subject simply and systematically, giving graduate students and practitioners a better understanding and enabling them to apply the methods in practice. The book develops Ito calculus and Fokker-Planck equations as parallel approaches to stochastic processes, using those methods in a unified way. The focus is on nonstationary processes, and statistical ensembles are emphasized in time series analysis. Stochastic calculus is developed using general martingales. Scaling and fat tails are presented via diffusive models. Fractional Brownian motion is thoroughly analyzed and contrasted with Ito processes. The Chapman-Kolmogorov and Fokker-Planck equations are shown in theory and by example to be more general than a Markov process. The book also presents new ideas in financial economics and a critical survey of econometrics.
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