|
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] |
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 * Local government, a form of public administration, usually the lowest tier of administration * Local news, coverage of events in a local context which would not normally be of interest to those of other localities * Local union, a locally based trade union organization which forms part of a larger union 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 * Local TV LLC, an American television broadcasting company * Locast, a non-profit streaming service offering local, over-the-air television * ''The Local'' (film), a 2008 action-drama film * '' The Local'', English-language news websites in several European countries Computing * .local, a network address component * Local variable, a variable that is given loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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". there have been 2232 volumes in this series. See also * ''Lecture Notes in Physics'' * ''Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' is a series of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Process
In mathematics, 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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...). 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 started at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. 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'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Informa ... References External links * Publications established in 1957 Mathematics journals University of Illinois Urbana-Champaign publications Quarterly journals English-language journ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, i.e. every finite linear combination of them is normally distributed. 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 distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 as first proposed by Kingman and proven by Iglehart and 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 and ::Z(t) = X(t) + R Y(t) with ''Y''(''t'') a ''d''–dimensional vector where * ''Y'' is continuous and non–decreasing with ''Y''(0)&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob–Meyer Decomposition Theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an 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 supermartingale Z is of Class D if Z_0=0 and the collection : \ is uniformly integrable.Protter (2005) The theorem Let Z be a cadlag supermartingale of class D. Then there exists a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Càdlàg
In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), 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 "continue à gauche, limite à droite", the left-right reversal of càdlàg, and càllàl for "continue à l'un, limite à l’autre" (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 be a metric space, and let . A function is called a càdlàg function if, for every , * the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ito Diffusion
Ito 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 * Princess Ito (died 861), Japanese imperial princess * Ito Giani (1941–2018), Italian sprinter * 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 (born 1995), Japanese fashion model and actress *Ito Smith (born 1995), American football player * Ito Curata (1959–2020), Filipino fashion designer * Ito Morabito (born 1977), French designer * Ito Ogawa (born 1973), Japanese novelist, lyricist, and translator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
