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List Of Things Named After Anatoliy Skorokhod
{{Short description, none These are things named after Anatoliy Skorokhod (1930-2011), a Ukrainian mathematician. Skorokhod * Skorokhod space * Skorokhod integral * Skorokhod problem Skorokhod's * Skorokhod's theorem: ** Skorokhod's embedding theorem ** Skorokhod's representation theorem In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a ... Skorokhod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatoliy Skorokhod
Anatoliy Volodymyrovych Skorokhod ( uk, Анато́лій Володи́мирович Скорохо́д; September 10, 1930January 3, 2011) was a USSR, Soviet and Ukraine, Ukrainian mathematician. Skorokhod is well-known for a comprehensive treatise on the theory of stochastic processes, co-authored with Iosif Gikhman, Gikhman. In the words of mathematician and probability theorist Daniel W. Stroock “Gikhman and Skorokhod have done an excellent job of presenting the theory in its present state of rich imperfection.” Career Skorokhod worked at Taras Shevchenko National University of Kyiv, Kyiv University from 1956 to 1964. He was subsequently at the Institute of Mathematics of the National Academy of Sciences of Ukraine from 1964 until 2002. Since 1993, he had been a professor at Michigan State University in the US, and a member of the American Academy of Arts and Sciences. He was an academician of the National Academy of Sciences of Ukraine from 1985 to his death in 2011. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod Space
Anatoliy Volodymyrovych Skorokhod ( uk, Анато́лій Володи́мирович Скорохо́д; September 10, 1930January 3, 2011) was a Soviet and Ukrainian mathematician. Skorokhod is well-known for a comprehensive treatise on the theory of stochastic processes, co-authored with Gikhman. In the words of mathematician and probability theorist Daniel W. Stroock “Gikhman and Skorokhod have done an excellent job of presenting the theory in its present state of rich imperfection.” Career Skorokhod worked at Kyiv University from 1956 to 1964. He was subsequently at the Institute of Mathematics of the National Academy of Sciences of Ukraine from 1964 until 2002. Since 1993, he had been a professor at Michigan State University in the US, and a member of the American Academy of Arts and Sciences. He was an academician of the National Academy of Sciences of Ukraine from 1985 to his death in 2011. His scientific works are on the theory of: * stochastic differential equa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod Integral
In mathematics, the Skorokhod integral (also named Hitsuda-Skorokhod integral), often denoted \delta, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts: * \delta is an extension of the Itô integral to non-adapted processes; * \delta is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus); * \delta is an infinite-dimensional generalization of the divergence operator from classical vector calculus. The integral was introduced by Hitsuda in 1972 and by Skorokhod in 1975. Definition Preliminaries: the Malliavin derivative Consider a fixed probability space (\Omega, \Sigma, \mathbf) and a Hilbert space H; \mathbf denotes expectation with respect to \mathbf \mathbf := \int_ X(\omega) \, \mathrm \mathbf(\omega). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod Problem
In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition. The problem is named after Anatoliy Skorokhod who first published the solution to a stochastic differential equation for a reflecting Brownian motion. Problem statement The classic version of the problem states that given a càdlàg process and an M-matrix ''R'', then stochastic processes and are said to solve the Skorokhod problem if for all non-negative ''t'' values, # ''W''(''t'') = ''X''(''t'') + ''R Z''(''t'') ≥ 0 # ''Z''(0) = 0 and d''Z''(''t'') ≥ 0 # \int_0^t W_i(s)\textZ_i(s)=0. The matrix ''R'' is often known as the reflection matrix, ''W''(''t'') as the reflected process and ''Z''(''t'') as the regulator process. See also List of things named after Anatoliy Skorokhod {{Short description, none These are things named after Anatoliy Skorokhod (1930-2011), a Ukrainian m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod's Embedding Theorem
In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process ( Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A. V. Skorokhod. Skorokhod's first embedding theorem Let ''X'' be a real-valued random variable with expected value 0 and finite variance; let ''W'' denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of ''W''), ''τ'', such that ''W''''τ'' has the same distribution as ''X'', :\operatornametau= \operatorname ^2/math> and :\operatorname tau^2\leq 4 \operatorname ^4 Skorokhod's second embedding theorem Let ''X''1, ''X''2, ... be a sequence of independent and identically distributed random variables In probability theory and statistics, a collection of random variables is independent and identica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod's Representation Theorem
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Soviet mathematician A. V. Skorokhod. Statement Let (\mu_n)_ be a sequence of probability measures on a metric space S such that \mu_n converges weakly to some probability measure \mu_\infty on S as n \to \infty. Suppose also that the support of \mu_\infty is separable. Then there exist S-valued random variables X_n defined on a common probability space (\Omega,\mathcal,\mathbf) such that the law of X_n is \mu_n for all n (including n=\infty) and such that (X_n)_ converges to X_\infty, \mathbf-almost surely. See also * Convergence in distribution In probability theory, there exist several different notions of convergence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |