|
Subordinator (mathematics)
In probability theory, a subordinator is a stochastic process that is non-negative and whose increments are stationary increments, stationary and independent increments, independent. Subordinators are a special class of Lévy process that play an important role in the theory of local time (mathematics), local time. In this context, subordinators describe the evolution of time within another stochastic process, the subordinated stochastic process. In other words, a subordinator will determine the random variable, random number of "time steps" that occur within the subordinated process for a given unit of chronological time. In order to be a subordinator a process must be a Lévy process It also must be increasing, almost surely, or an additive process. Definition A subordinator is a real-valued stochastic process X=(X_t)_ that is a non-negative and a Lévy process. Subordinators are the stochastic processes X=(X_t)_ that have all of the following properties: * X_0=0 almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance Gamma Process
In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma (VG) process, also known as Laplace motion, is a Lévy process determined by a random time change. The process has finite moments, distinguishing it from many Lévy processes. There is no diffusion component in the VG process and it is thus a pure jump process. The increments are independent and follow a variance-gamma distribution, which is a generalization of the Laplace distribution. There are several representations of the VG process that relate it to other processes. It can for example be written as a Brownian motion W(t) with drift \theta t subjected to a random time change which follows a gamma process \Gamma(t; 1, \nu) (equivalently one finds in literature the notation \Gamma(t;\gamma=1/\nu,\lambda=1/\nu)): : X^(t; \sigma, \nu, \theta) \;:=\; \theta \,\Gamma(t; 1, \nu) + \sigma\,W(\Gamma(t; 1, \nu)) \quad. An alternative way of stating this is that the variance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intensity Measure
In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure. Definition Let \zeta be a random measure on the measurable space (S, \mathcal A) and denote the expected value of a random element Y with \operatorname E . The intensity measure : \operatorname E \zeta \colon \mathcal A \to ,\infty of \zeta is defined as : \operatorname E \zeta(A)= \operatorname E zeta(A) for all A \in \mathcal A. Note the difference in notation between the expectation value of a random element Y , denoted by \operatorname E and the intensity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points randomly located on a Space (mathematics), mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile."van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, , and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995/ref> It is a special case of the inverse-gamma distribution. It is a stable distribution. Definition The probability density function of the Lévy distribution over the domain x \ge \mu is : f(x; \mu, c) = \sqrt \, \frac, where \mu is the location parameter, and c is the scale parameter. The cumulative distribution function is : F(x; \mu, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Process
In probability theory, a Cauchy process is a type of stochastic process. There are symmetric and asymmetry, asymmetric forms of the Cauchy process. The unspecified term "Cauchy process" is often used to refer to the symmetric Cauchy process. The Cauchy process has a number of properties: #It is a Lévy process #It is a stable process #It is a pure jump process #Its moment (mathematics), moments are infinity, infinite. Symmetric Cauchy process The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévy distribution, Lévy subordinator (mathematics), subordinator. The Lévy subordinator is a process associated with a Lévy distribution having location parameter of 0 and a scale parameter of t^2/2. The Lévy distribution is a special case of the inverse-gamma distribution. So, using C to represent the Cauchy process and L to represent the Lévy subordinator, the symmetric Cauchy process can be described as: : C(t; 0, 1) \;:=\;W(L( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary increments, stationary independent increments). It occurs frequently in pure and applied mathematics, economy, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingale (probability theory), martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Process
A gamma process, also called the ''Moran-Gamma subordinator'', is a two-parameter stochastic process which models the accumulation of ''effort'' or ''wear'' over time. The gamma process has independent and stationary increments which follow the gamma distribution, hence the name. The gamma process is studied in mathematics, statistics, probability theory, and stochastics, with particular applications in deterioration modeling and mathematical finance. Notation The gamma process is often abbreviated as X(t)\equiv\Gamma(t;\gamma, \lambda) where t represents the time from 0. The shape parameter \gamma (inversely) controls the jump size, and the rate parameter \lambda controls the rate of jump arrivals, analogously with the gamma distribution. Both \gamma and \lambda must be greater than 0. We use the gamma function and gamma distribution in this article, so the reader should distinguish between \Gamma(\cdot) (the gamma function), \Gamma(\gamma, \lambda) (the gamma distribution), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of Randomness, random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall Linear momentum, linear and Angular momentum, angular momenta remain null over time. The Kinetic energy, kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's in ... [...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] |
