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Itô Diffusion
In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô. Overview A (time-homogeneous) Itô diffusion in ''n''-dimensional Euclidean space R''n'' is a process ''X'' : , +∞) × Ω → R''n'' defined on a probability space (Ω, Σ, P) and satisfying a stochastic differential equation of the form :\mathrm X_ = b(X_t) \, \mathrm t + \sigma (X_) \, \mathrm B_, where ''B'' is an ''m''-dimensional Brownian motion and ''b'' : R''n'' → R''n'' and σ : R''n'' → R''n''×''m'' satisfy the usual Lipschitz continuity condition :, b(x) - b(y) , + , \sigma (x) - \sigma (y) , \leq C ... [...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] |
Stochastic Drift
In probability theory, stochastic drift is the change of the average value of a stochastic (random) process. A related concept is the drift rate, which is the rate at which the average changes. For example, a process that counts the number of heads in a series of n fair coin tosses has a drift rate of 1/2 per toss. This is in contrast to the random fluctuations about this average value. The stochastic mean of that coin-toss process is 1/2 and the drift rate of the stochastic mean is 0, assuming 1 = heads and 0 = tails. Stochastic drifts in population studies Longitudinal studies of secular events are frequently conceptualized as consisting of a trend component fitted by a polynomial, a cyclical component often fitted by an analysis based on autocorrelations or on a Fourier series, and a random component (stochastic drift) to be removed. In the course of the time series analysis, identification of cyclical and stochastic drift components is often attempted by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only exception). * Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin's Formula
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin. Statement of the theorem Let ''X'' be the R''n''-valued Itō diffusion solving the stochastic differential equation :\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_. For a point ''x'' ∈ R''n'', let P''x'' denote the law of ''X'' given initial datum ''X''0 = ''x'', and let E''x'' denote expectation with respect to P''x''. Let ''A'' be the infinitesimal generator of ''X'', defined by its action on compactly-supported ''C''2 (twice differentiable with continuous second derivative) functions ''f'' : R''n'' → R as :A f (x) = \lim_ \frac or, equivalently, :A f (x) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Characteristic Operator
''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Generator (stochastic Processes)
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation (which describes the evolution of statistics of the process); its ''L''2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation (which describes the evolution of the probability density functions of the process). Definition General case For a Feller process (X_t)_ with Feller semigroup T=(T_t)_ and state space E we define the generator (A,D(A)) by :D(A)=\left\, :A f=\lim_ \frac, for any f\in D(A). Here C_(E) denotes the Banach space of continuous functions on E vanishing at infinity, equipped with the supremum norm, and T_t f(x)= \mathbb^x f(X_t)=\mathbb(f(X_t), X_0=x). In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Markov Property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov assumption is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov assumption is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller-continuous Process
In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller. Definition Let ''X'' : bounded, continuous and Σ-measurable function ''g'' : R''n'' → R, E''x'' 'g''(''X''''t'')depends continuously upon ''x''. Examples * Every process ''X'' whose paths are almost surely constant for all time is a Feller-continuous process, since then E''x'' 'g''(''X''''t'')is simply ''g''(''x''), which, by hypothesis, depends continuously upon ''x''. * Every Itô diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process. See also * Continuous stochastic process In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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