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trudinger's theorem | In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space n... | wikipedia |
trudinger's theorem | The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem: Let Ω {\displaystyle \Omega } be a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} satisfying the cone condition. Let m p = n {\displaystyle mp=n} and p > 1 {\displaystyle p>1} . Set A ( t ) = exp ( t n / ( n − m... | wikipedia |
trudinger's theorem | {\displaystyle A(t)=\exp \left(t^{n/(n-m)}\right)-1.} Then there exists the embedding W m , p ( Ω ) ↪ L A ( Ω ) {\displaystyle W^{m,p}(\Omega )\hookrightarrow L_{A}(\Omega )} where L A ( Ω ) = { u ∈ M f ( Ω ): ‖ u ‖ A , Ω = inf { k > 0: ∫ Ω A ( | u ( x ) | k ) d x ≤ 1 } < ∞ } . {\displaystyle L_{A}(\Omega )=\left\{u\in... | wikipedia |
wiener's tauberian theorem | In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can be approximated by linear combinations of translations of a gi... | wikipedia |
zorich's theorem | In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938. | wikipedia |
besicovitch covering theorem | In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant cN depending only on the d... | wikipedia |
besicovitch covering theorem | Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from ... | wikipedia |
carathéodory function | In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function wi... | wikipedia |
hermitian function | In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)} (where the ∗ {\displaystyle ^{*}} indicates the complex conjugate) for all x {\displ... | wikipedia |
equicontinuous linear maps | In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the fo... | wikipedia |
equicontinuous linear maps | As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact spa... | wikipedia |
function of bounded variation | In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means tha... | wikipedia |
function of bounded variation | Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many. | wikipedia |
function of bounded variation | In the case of several variables, a function f defined on an open subset Ω of R n {\displaystyle \mathbb {R} ^{n}} is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of bounded variation is that they form an algebra ... | wikipedia |
complete metric space | In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, becau... | wikipedia |
thin set (analysis) | In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the set on which a holomorphic function vanishes is closed and has empty interior (by the Identity theorem), a thin set is ... | wikipedia |
function of a real variable | In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. Most r... | wikipedia |
function of a real variable | However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers or the qua... | wikipedia |
function of a real variable | If the codomain has a structure of R {\displaystyle \mathbb {R} } -algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real va... | wikipedia |
continuous functions on a compact hausdorff space | In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted by C ( X ) , {\displaystyle {\mathcal {C}}(X),} is a vector space with respect ... | wikipedia |
continuous functions on a compact hausdorff space | {\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm. (Rudin 1973, §11.3) | wikipedia |
function (mathematics) | In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact s... | wikipedia |
asymptotic theory | In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant comp... | wikipedia |
asymptotic theory | This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less ... | wikipedia |
constructive function theory | In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein. | wikipedia |
power series ring | In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain. Let f = ∑ a n X n ∈ R ] , {\displaystyle f=\sum a_{n}X^{n}\... | wikipedia |
power series ring | {\displaystyle f(x)=\sum _{n\geq 0}a_{n}x^{n}.} This series is guaranteed to converge in S {\displaystyle S} given the above assumptions on x {\displaystyle x} . Furthermore, we have ( f + g ) ( x ) = f ( x ) + g ( x ) {\displaystyle (f+g)(x)=f(x)+g(x)} and ( f g ) ( x ) = f ( x ) g ( x ) . | wikipedia |
power series ring | {\displaystyle (fg)(x)=f(x)g(x).} Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved. Since the topology on R ] {\displaystyle R]} is the ( X ) {\displaystyle (X)} -adic topology and R ] {\displaystyle R]} is complete, we can in particular apply power series to other p... | wikipedia |
power series ring | {\displaystyle f\in R].} With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f {\displaystyle f} whose constant coefficient a = f ( 0 ) {\displaystyle a=f(0)} is invertible in R {\displaystyle R}: f − 1 = ∑ n ≥ 0 a − n − 1 ( a − f ) n . {\displaystyle f^{-1}=\sum _{n\ge... | wikipedia |
idempotent analysis | In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕ A = A {\displaystyle A\oplus A=A} . == References == | wikipedia |
proper convex function | In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value − ∞ {\displaystyle -\infty } and also is not identically equal to + ∞ . {\displaystyle +\infty .} In conve... | wikipedia |
proper convex function | Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the function. If the function takes − ∞ {\displaystyle -\infty } as a value then − ∞ {\displaystyle -\infty } is necessarily the global minimum value and the minimization prob... | wikipedia |
proper convex function | Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases. If the p... | wikipedia |
parameter | In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form F ( t ) = ∫ x 0 ( t ) x 1 ( t ) f ( x ; t ) d x . {\displaystyle F(t)=\int _{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.} In this formula, t is the argument of the function F, and on the right-hand side the parameter on which t... | wikipedia |
parameter | When evaluating the integral, t is held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t, we then consider t to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration... | wikipedia |
microlocalization functor | In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier ... | wikipedia |
alexandrov theorem | In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f: U → R m {\displaystyle f\colon U\to \mathbb {R} ^{m}} is a convex function, then f {\displaystyle f} has a second derivative almost everywhere... | wikipedia |
bessel–clifford function | In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If π ( x ) = 1 Π ( x ) = 1 Γ ( x + 1 ) {\displaystyle \pi (x)={\frac... | wikipedia |
bohr–mollerup theorem | In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t} as the o... | wikipedia |
haar integral | In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1... | wikipedia |
karamata's tauberian theorem | In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equi... | wikipedia |
karamata's tauberian theorem | The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. : 226 In 1930, Jovan Karamata gave a new and much simpler proof. : 226 | wikipedia |
hilbert–schmidt theorem | In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems. | wikipedia |
kakutani fixed-point theorem | In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixe... | wikipedia |
kakutani fixed-point theorem | Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. | wikipedia |
rademacher–menchov theorem | In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere. | wikipedia |
szegő limit theorems | In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő. | wikipedia |
weierstrass approximation theorem | In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials,... | wikipedia |
weierstrass approximation theorem | His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of... | wikipedia |
mean-periodic function | In mathematical analysis, the concept of a mean-periodic function is a generalization of the concept of a periodic function introduced in 1935 by Jean Delsarte. Further results were made by Laurent Schwartz. | wikipedia |
final value theorem | In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s... | wikipedia |
initial value theorem | In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaysty... | wikipedia |
intermediate value theorem | In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval , then it takes on any given value between f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} at some point within the interval. This has two important corol... | wikipedia |
mean value theorem for divided differences | In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. | wikipedia |
differentiability class | In mathematical analysis, the smoothness of a function is a property measured by the number of continuous 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... | wikipedia |
spaces of test functions and distributions | In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset U ⊆ R n {\di... | wikipedia |
spaces of test functions and distributions | There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If U = R n {\displaystyle U=\mathbb {R} ^{n}} then the use of Schwartz functions as test functions gives rise to a certain subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} whose elem... | wikipedia |
spaces of test functions and distributions | The set of tempered distributions forms a vector subspace of the space of distributions D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of ... | wikipedia |
universal chord theorem | In mathematical analysis, the universal chord theorem states that if a function f is continuous on and satisfies f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} , then for every natural number n {\displaystyle n} , there exists some x ∈ {\displaystyle x\in } such that f ( x ) = f ( x + b − a n ) {\displaystyle f(x)=f\lef... | wikipedia |
multimedia | In mathematical and scientific research, multimedia is mainly used for modeling and simulation. For example, a scientist can look at a molecular model of a particular substance and manipulate it to arrive at a new substance. Representative research can be found in journals such as the Journal of Multimedia. | wikipedia |
multimedia | One well known example of this being applied would be in the movie Interstellar where Executive Director Kip Thorne helped create one of the most realistic depictions of a blackhole in film. The visual effects team under Paul Franklin took Kip Thorne's mathematical data and applied it into their own visual effects engi... | wikipedia |
radó's theorem (riemann surfaces) | In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface... | wikipedia |
schottky theorem | In mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0). Schottky's original theore... | wikipedia |
geometric function theory | In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | wikipedia |
semi colon | In the calculus of relations, the semicolon is used in infix notation for the composition of relations: A ; B = { ( x , z ): ∃ y x A y ∧ y B z } . {\displaystyle A;B\ =\ \{(x,z):\exists y\ \ xAy\ \land \ yBz\}~.} The ; Humphrey point is sometimes used as the "decimal point" in duodecimal numbers: 54;612 equals 64.510. | wikipedia |
topkis's theorem | In mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes. The result states that if f is supermodular in (x,θ), and D is a lattic... | wikipedia |
isoelastic function | In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent v... | wikipedia |
arrow–debreu model | In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity i... | wikipedia |
arrow–debreu model | In general, there may be many equilibria. Arrow (1972) and Debreu (1983) were separately awarded the Nobel Prize in Economics for their development of the model. McKenzie however was not awarded. | wikipedia |
solomon mikhlin | In mathematical elasticity theory, Mikhlin was concerned by three themes: the plane problem (mainly from 1932 to 1935), the theory of shells (from 1954) and the Cosserat spectrum (from 1967 to 1973). Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains. The f... | wikipedia |
solomon mikhlin | Mikhlin studied its convergence and gave applications to special applied problems. He proved existence theorems for the fundamental problems of plane elasticity involving inhomogeneous anisotropic media: these results are collected in the book (Mikhlin 1957). Concerning the theory of shells, there are several Mikhlin's... | wikipedia |
solomon mikhlin | He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress. As a result of his study of this problem, Mikhlin also gave a new (invariant) form of the basic equations of the theory. He also proved a theo... | wikipedia |
solomon mikhlin | Mikhlin studied also the spectrum of the operator pencil of the classical linear elastostatic operator or Navier–Cauchy operator A ( ω ) u = Δ 2 u + ω ∇ ( ∇ ⋅ u ) {\displaystyle {\boldsymbol {\mathcal {A}}}(\omega ){\boldsymbol {u}}=\Delta _{2}{\boldsymbol {u}}+\omega \nabla \left(\nabla \cdot {\boldsymbol {u}}\right)}... | wikipedia |
littlewood–offord problem | In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subse... | wikipedia |
littlewood–offord problem | This bound is sharp; equality is attained when all vectors in S are equal. In 1966, Kleitman showed that the same bound held for complex numbers. In 1970, he extended this to the setting when V is a normed space.Suppose S = {v1, …, vn}. | wikipedia |
littlewood–offord problem | By subtracting 1 2 ∑ i = 1 n v i {\displaystyle {\frac {1}{2}}\sum _{i=1}^{n}v_{i}} from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form ∑ i = 1 n ε i v i {\displaystyle \su... | wikipedia |
smoluchowski equation | In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of th... | wikipedia |
monte carlo option model | In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how... | wikipedia |
black-scholes equation | In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.... | wikipedia |
constant elasticity of variance model | In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by J... | wikipedia |
cox–ingersoll–ross model | In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was in... | wikipedia |
doob decomposition theorem | In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let X = (X0, X1, . . . | wikipedia |
doob decomposition theorem | , XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, . . . | wikipedia |
doob decomposition theorem | , FN), and let Q {\displaystyle \mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . . | wikipedia |
doob decomposition theorem | . , UN) denote the Snell envelope of X with respect to Q {\displaystyle \mathbb {Q} } . The Snell envelope is the smallest Q {\displaystyle \mathbb {Q} } -supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. | wikipedia |
doob decomposition theorem | Let U = M + A denote the Doob decomposition with respect to Q {\displaystyle \mathbb {Q} } of the Snell envelope U into a martingale M = (M0, M1, . . . | wikipedia |
doob decomposition theorem | , MN) and a decreasing predictable process A = (A0, A1, . . . | wikipedia |
doob decomposition theorem | , AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way is τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0. {\displaystyle \tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<... | wikipedia |
doob decomposition theorem | . . , N − 1}, hence τmax is indeed a stopping time. It gives the last moment before the discounted value of the American option will drop in expectation; up to time τmax the discounted value process U is a martingale with respect to Q {\displaystyle \mathbb {Q} } . | wikipedia |
sabr volatility model | In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especial... | wikipedia |
local volatility | In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form d S t = ( r t − d t ) S t d t + σ t S t d W t {\displaystyle dS_{t}=(r_{t}-d_{t})S_{t}\,dt+\sigma _{t}S_{t}\,dW_{t}} ,under the risk neutral measure, where r t {\disp... | wikipedia |
local volatility | And when such volatility is merely a function of the current underlying asset level St and of time t, we have a local volatility model. The local volatility model is a useful simplification of the stochastic volatility model. | wikipedia |
local volatility | "Local volatility" is thus a term used in quantitative finance to denote the set of diffusion coefficients, σ t = σ ( S t , t ) {\displaystyle \sigma _{t}=\sigma (S_{t},t)} , that are consistent with market prices for all options on a given underlying, yielding an asset price model of the type d S t = ( r t − d t ) S t... | wikipedia |
thompson uniqueness theorem | In mathematical finite group theory, Thompson's original uniqueness theorem (Feit & Thompson 1963, theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof of the uniq... | wikipedia |
n-group (finite group theory) | In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups. | wikipedia |
baer–suzuki theorem | In mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements of the conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a s... | wikipedia |
brauer–fowler theorem | In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another resul... | wikipedia |
gorenstein–harada theorem | In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973, 1974) in a 464-page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups.Finite simple groups of section 2 that rank at least 5, have S... | wikipedia |
l-balance theorem | In mathematical finite group theory, the L-balance theorem was proved by Gorenstein & Walter (1975). The letter L stands for the layer of a group, and "balance" refers to the property discussed below. | wikipedia |
thompson transitivity theorem | In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson un... | wikipedia |
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