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Russo–Vallois Integral
In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral :\int f \, dg=\int fg' \, ds for suitable functions f and g. The idea is to replace the derivative g' by the difference quotient :g(s+\varepsilon)-g(s)\over\varepsilon and to pull the limit out of the integral. In addition one changes the type of convergence. Definitions Definition: A sequence H_n of stochastic processes Convergence of random variables, converges uniformly on compact sets in probability to a process H, :H=\text\lim_H_n, if, for every \varepsilon>0 and T>0, :\lim_\mathbb(\sup_, H_n(t)-H(t), >\varepsilon)=0. One sets: :I^-(\varepsilon,t,f,dg)=\int_0^tf(s)(g(s+\varepsilon)-g(s))\,ds :I^+(\varepsilon,t,f,dg)=\int_0^t f(s)(g(s)-g(s-\varepsilon)) \, ds and :[f,g]_\varepsilon (t)=\int_0^t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s))\,ds. Definition: The forward integral is defined as the ucp-limit of :I^-: \int_0^t fd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Stieltjes Integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability. Formal definition The Riemann–Stieltjes integral of a real-valued function f of a real variable on the interval ,b/math> with respect to another real-to-real function g is denoted by :\int_^b f(x) \, \mathrmg(x). Its definition uses a sequence of partitions P of the interval ,b/math> :P=\. The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches 0 , of the approximating sum :S(P,f,g) = \sum_^ f(c_i)\left g(x_) - g(x_) \right/math> where c_i is in the i-th subinterval _,x_/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Random Variables
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution. The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semimartingale
In probability theory, a real-valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales. Definition A real-valued process ''X'' defined on the filtered probability space (Ω,''F'',(''F''''t'')''t'' ≥ 0,P) is called a semimartingale if it can be decomposed as :X_t = M_t + A_t where ''M'' is a local martingale and ''A'' is a càdlàg adapted process of locally bounded variation. This means that for almost all \omega \in \Omega and all compact intervals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Itô Integral
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 designe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Variation Process
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued stochastic process defined on a probability space (\Omega,\mathcal,\mathbb) and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as t, defined as : t=\lim_\sum_^n(X_-X_)^2 where P ranges over partitions of the interval ,t/math> and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian motion. More generally, the covariation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ito Formula
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] |
Triebel
Triebel is a municipality in the Vogtlandkreis district, in Saxony, Germany, situated south of the town of Plauen Plauen (; ; ) is a town in Saxony, Germany with a population of around 65,000. It is Saxony's 5th most populated city after Leipzig, Dresden, Chemnitz and Zwickau, the second-largest city of the Vogtland after Gera, as well as the largest cit .... Fotographs Triebel Kirchberg 2010.jpg, Kirchberg in Triebel Triebel Wiederaufbau Kirche 2014.jpg, Saint Ägidien Church (2014) References Municipalities in Saxony Vogtlandkreis {{Vogtlandkreis-geo-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
