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Local Time (mathematics)
In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields. Formal definition For a continuous real-valued semimartingale (B_s)_, the local time of B at the point x is the stochastic process which is informally defined by :L^x(t) =\int_0^t \delta(x-B_s)\,d s, where \delta is the Dirac delta function and /math> is the quadratic variation. It is a notion invented by Paul Lévy. The basic idea is that L^x(t) is an (appropriately rescaled and time-parametrized) measure of how much time B_s has spent at x up to time t. More rigorously, it may be written as the almost sure limit : L^x(t) =\lim_ \frac \int_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Local Times Surface
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * ''Local'' (novel), a 2001 novel by Jaideep Varma * ''The Local'' (film), a 2008 action-drama film * ''The Local'', English-language news websites in several European countries Computing * .local, a network address component Mathematics * Local property, a property which occurs on ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points * Local ring, type of ring in commutative algebra Other uses * Pub, a drinking establishment, known as a "local" to its regulars See also * * * Local group (other) * Locale (other) * Localism (other) * Locality (other) * Localization (other) * Locus (other) * Lokal (other) Lokal may refer to: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Lecture Notes In Mathematics
''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Springer Science+Business Media (formerly Springer-Verlag). The intent of the series is to publish not only lecture notes, but results from seminars and conferences, more quickly than the several-years-long process of publishing polished journal papers in mathematics. In order to speed the publication process, early volumes of the series (before electronic publishing) were reproduced photographically from typewritten manuscripts. According to Earl Taft, it has been "enormously successful" and "is considered a very valuable service to the mathematical community". As of 2023, there has been over 2300 volumes in the series. See also * ''Lecture Notes in Physics'' * ''Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Random Field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold. Formal definition Given a probability space (\Omega, \mathcal, P), an ''X''-valued random field is a collection of ''X''-valued random variables indexed by elements in a topological space ''T''. That is, a random field ''F'' is a collection : \ where each F_t is an ''X''-valued random variable. Examples In its discrete version, a random field is a list of ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Bessel Process
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., a Bessel process, named after Friedrich Bessel, is a type of stochastic process. Formal definition The Bessel process of order ''n'' is the real-valued process ''X'' given (when ''n'' ≥ 2) by :X_t = \, W_t \, , where , , ·, , denotes the Euclidean norm in R''n'' and ''W'' is an ''n''-dimensional Wiener process ( Brownian motion). For any ''n'', the ''n''-dimensional Bessel process is the solution to the stochastic differential equation (SDE) :dX_t = dW_t + \frac\frac where W is a 1-dimensional Wiener process ( Brownian motion). Note that this SDE makes sense for any real parameter n (although the drift term is singular at zero). Notation A notation for the Bessel process of dimension start ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Illinois Journal Of Mathematics
The ''Illinois Journal of Mathematics'' is a quarterly peer-reviewed scientific journal of mathematics published by Duke University Press on behalf of the University of Illinois. It was established in 1957 by Reinhold Baer, Joseph L. Doob, Abraham Taub, George W. Whitehead, and Oscar Zariski. The journal published the proof of the four color theorem by Kenneth Appel and Wolfgang Haken, which featured a then-unusual tabulation of computer-generated cases. Abstracting and indexing The journal is indexed and abstracted in: *MathSciNet *Scopus *zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ... References External links * Academic journals established in 1957 Mathematics journals University of Illinois Urbana-Champaign publications Quarterly journals English-language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Reflecting Brownian Motion
In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. In the Physics, physical literature, this process describes diffusion in a confined space and it is often called confined Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls. RBMs have been shown to describe queueing models experiencing heavy traffic approximation, heavy traffic as first proposed by John Kingman, Kingman and proven by Iglehart and Ward Whitt, Whitt. Definition A ''d''–dimensional reflected Brownian motion ''Z'' is a stochastic process on \mathbb R^d_+ uniquely defined by * a ''d''–dimensional drift vector ''μ'' * a ''d''×''d'' non-singular covariance matrix ''Σ'' and * a ''d''×''d'' reflection matrix ''R''. where ''X''(''t'') is an unconstrained Brownian motion with drift ''μ'' and variance ''Σ'', and ::Z(t) = X(t) + R Y(t) with ''Y' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Doob–Meyer Decomposition Theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a Martingale (probability theory)#Submartingales and supermartingales, submartingale may be decomposed in a unique way as the sum of a Martingale (probability theory), martingale and an increasing process, increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer. History In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 Paul-André Meyer proved such a theorem, which became known as the Doob-Meyer decomposition. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.Protter 2005 Class D supermartingales A càdlàg Martingale_(probability_theory)#Submartingales_and_supermartingales, supe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "", the left-right reversal of càdlàg, and càllàl for "" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let (M, d) be a metric space, and let E \subseteq \mathbb. A function f:E \to M is called a càdlàg function if, for every t \in E, * the left limit f(t-) := \lim_f(s) exists; and * the right limit f(t+) := \lim_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ito Diffusion
Ito, Itō or Itoh may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan * Itō, Shizuoka People * Itō (surname), for people with the Japanese surname Itō * , Japanese voice actor * Kiyosi Itô (1915–2008), Japanese mathematician * Princess Ito (died 861), Japanese imperial princess * Ito Giani (1941–2018), Italian sprinter * Ito (footballer, born 1961), full name Andrés Alonso García, Spanish footballer * Ito (footballer, born 1975), full name Antonio Álvarez Pérez, Spanish footballer * Ito (footballer, born 1992), full name Jorge Delgado Fidalgo, Spanish footballer * Ito (footballer, born 1994), full name Mario Manuel de Oliveira, Angolan footballer * , Japanese fashion model and actress *Ito Smith (born 1995), American football player * Ito Curata (1959–2020), Filipino fashion designer * It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |