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Fractional Brownian Motion
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process ''BH''(''t'') on , ''T'' that starts at zero, has expectation zero for all ''t'' in , ''T'' and has the following covariance function: :E _H(t) B_H (s)\tfrac (, t, ^+, s, ^-, t-s, ^), where ''H'' is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by . The value of ''H'' determines what kind of process the ''fBm'' is: * if ''H'' = 1/2 then the process is in fact a Brownian motion or Wiener process; * if ''H'' > 1/2 then the increments of the process are positively correlated; * if ''H'' < 1/2 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Integration
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by :\mathbb^q f is the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' So, \frac = \mathcal^\left\ which generalizes to \mathbb^qf(t) = \mathcal^\left\. Under the bilateral Laplace transform, here denoted by \mathcal and defined as \mathcal (t)=\int_^\infty e^ f(t)\, dt, differentiation transforms into a multiplication \mathcal\left frac\right= s\mathcal (t) Generalizing to arbitrary order and solving for \mathbb^qf(t), one obtains \mathbb^qf(t)=\mathcal^\left\. Representation via Newton series is the Newton interpolation over consecutive integer orders: \mathbb^qf(t) =\sum_^ \binom m \sum_^m\binom mk(-1)^f^(x). For fractional derivative definitions described in this section, the following identities hold: :\ ... [...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. An R''n''-valued process ''X'' = (''X''1,…,''X''''n'') is a semimartingale i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ito Calculus
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 transla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Dimension
A box (plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and can be used for a variety of purposes from functional to decorative. Boxes may be made of a variety of materials, both durable, such as wood and metal; and non-durable, such as corrugated fiberboard and paperboard. Corrugated metal boxes are commonly used as shipping containers. Most commonly, boxes have flat, parallel, rectangular sides, making them rectangular prisms; but boxes may also have other shapes. Rectangular prisms are often referred to colloquially as "boxes." Boxes may be closed and shut with flaps, doors, or a separate lid. They can be secured shut with adhesives, tapes, or more decorative or elaborately functional mechanisms, such as a catch, fastener, clasp or lock (security device), lock. Types Packaging Several types of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that : , f(x) - f(y) , \leq C\, x - y\, ^ for all ''x'' and ''y'' in the domain of ''f''. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: : Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous ⊂ continuous, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nowhere Differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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