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Malliavin Calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, J-M. Bismut, Shinzo Watanabe, I. Shigekawa, and so on finally completed the foundations. Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well. The cal ... [...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] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often with the help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence (measure Theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Definition Let \mu and \nu be two measures on the measurable space (X, \mathcal A), and let \mathcal_\mu := \ and \mathcal_\nu := \ be the sets of \mu- null sets and \nu-null sets, respectively. Then the measure \nu is said to be absolutely continuous in reference to \mu if and only if \mathcal N_\nu \supseteq \mathcal N_\mu. This is denoted as \nu \ll \mu. The two measures are called equivalent if and only if \mu \ll \nu and \nu \ll \mu, which is denoted as \mu \sim \nu. That is, two measures are equivalent if they satisfy \mathcal N_\mu = \mathcal N_\nu. Examples On the real line Define the two measures on the real line as \mu(A)= \int_A \mathbf 1_(x) \mathrm dx \nu(A)= \int_A x^2 \mathbf 1_(x) \mathrm dx for all Borel sets A. Then \mu and \nu are equivalent, since all sets outside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Space
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is an RKHS if the point-evaluation functional L_x:H\to\mathbb, L_x(f)=f(x), is continuous for every x\in X. Equivalently, H is an RKHS if there exists a function K_x \in H such that, for all f \in H,\langle f, K_x \rangle = f(x).The function K_x is then called the ''reproducing kernel'', and it reproduces the value of f at x via the inner product. An immediate consequence of this property is that convergence in norm implies uniform convergence on any subset of X on which \, K_x\, is bounded. However, the converse does not necessarily hold. Often the set X carries a topology, and \, K_x\, depends continuously on x\in X, in which case: convergence in norm implies uniform convergence on compact subsets of X. It is not entirely straightforwar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Operator
In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product \langle \cdot,\cdot\rangle , the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by :\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, \mathrm \mathbf (z) for all ''x'' and ''y'' in ''H''. The covariance operator ''C'' is then defined by :\mathrm(x, y) = \langle Cx, y \rangle (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint. Even more generally, for a probability measure P on a Banach space ''B'', the covariance of P is the bilinear form on the algebraic dual ''B''#, defined by :\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, \mathrm \mathbf (z) where \langle x, z \rangle is now the value of the linear functional ''x'' on the element ''z''. Quite similarly, the covariance function o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex Vector Space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cameron–Martin Theorem
In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space. Motivation The standard Gaussian measure \gamma^n on n-dimensional Euclidean space \mathbf^n is not translation- invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional Lebesgue measure, denoted here dx.) Instead, a measurable subset A has Gaussian measure :\gamma_n(A) = \frac\int_A \exp\left(-\tfrac12\langle x, x\rangle_\right)\,dx. Here \langle x,x\rangle_ refers to the standard Euclidean dot product in \mathbf^n. The Gaussian measure of the translation of A by a vector h \in \mathbf^n is :\begin \gamma_n(A-h) &= \frac\int_A \exp\left(-\tfrac12\langle x-h, x-h\rangle_\right)\,dx\\ pt&=\frac\int_A \exp\left(\frac\right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irving Segal
Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for his developments in quantum field theory and in functional and harmonic analysis, in particular his innovation of the algebraic axioms known as C*-algebra. Biography Irving Ezra Segal was born in the Bronx on September 13, 1918, to Jewish parents. He attended school in Trenton. In 1934 he was admitted to Princeton University, at the age of 16. He was elected to Phi Beta Kappa, completed his undergraduate studies in just three years time, graduated with highest honors with a bachelor's degree in 1937, and was awarded the George B. Covington Prize in Mathematics. He was then admitted to Yale, and in another three years time had completed his doctorate, receiving his Doctor of Philosophy degree in 1940. Segal taught at Harvard University, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space (mathematics)
In mathematics, a space is a set (sometimes known as a ''universe'') endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Probability Space
In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables. Definition A Gaussian probability space (\Omega,\mathcal,P,\mathcal,\mathcal^_) consists of * a (complete) probability space (\Omega,\mathcal,P), * a closed linear subspace \mathcal\subset L^2(\Omega,\mathcal,P) called the ''Gaussian space'' such that all X\in \mathcal are mean zero Gaussian variables. Their σ-algebra is denoted as \mathcal_. * a σ-algebra \mathcal^_ called the ''transverse σ-algebra'' which is defined through :: \mathcal=\mathcal_ \otimes \mathcal^_. Irreducibility A Gaussian probability space is called ''irreducible'' if \mathcal=\mathcal_. Such spaces are denoted as (\Omega,\mathcal,P,\mathcal). Non-irreducible spaces are used to work on sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gateaux Derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics. Unlike other forms of derivatives, the Gateaux differential of a function may be a nonlinear operator. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
