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Clark–Ocone Theorem
In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function ''F'' defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itô integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978). Statement of the theorem Let ''C''0( , ''T'' R) (or simply ''C''0 for short) be classical Wiener space with Wiener measure ''γ''. Let ''F'' : ''C''0 → R be a BC1 function, i.e. ''F'' is bounded and Fréchet differentiable with bounded derivative D''F'' : ''C''0 → Lin(''C''0; R). Then :F(\sigma) = \int_ F(p) \, \mathrm \gamma(p) + \int_^ \mathbf \left \Sigma_ \right(\sigma) \, \mathrm \sigma_. In the above * ''F''(''σ'') is the value of the function ''F' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts Operator
In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications. Definition Let ''E'' be a Banach space such that both ''E'' and its continuous dual space ''E''∗ are separable spaces; let ''μ'' be a Borel measure on ''E''. Let ''S'' be any (fixed) subset of the class of functions defined on ''E''. A linear operator ''A'' : ''S'' → ''L''2(''E'', ''μ''; R) is said to be an integration by parts operator for ''μ'' if :\int_ \mathrm \varphi(x) h(x) \, \mathrm \mu(x) = \int_ \varphi(x) (A h)(x) \, \mathrm \mu(x) for every ''C''1 function ''φ'' : ''E'' → R and all ''h'' ∈ ''S'' for which either side of the above equality makes sense. In the above, D''φ''(''x'') denotes the Fréchet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Representation Theorem For Classical Wiener Space
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion. The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using Malliavin calculus. Similar theorems also exist for martingales on filtrations induced by jump processes, for example, by Markov chains. Statement Let B_t be a Brownian motion on a standard filtered probability space (\Omega, \mathcal,\mathcal_t, P ) and let \mathcal_t be the augmented filtration generated by B. If ''X'' is a square integrable random variable measurable with respect to \mathcal_\infty, then there exists a predictable process ''C'' which is adapted In biology, adaptation has three related meanings. Firstly, it is the dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, Bismut, S. 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 calculus allows ... [...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] |
Adapted Process
In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every realisation and every ''n'', ''Xn'' is known at time ''n''. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * I be an index set with a total order \leq (often, I is \mathbb, \mathbb_0, , T/math> or filtration of the sigma algebra \mathcal; * (S,\Sigma) be a measurable space, the ''state space''; * X: I \times \Omega \to S be a stochastic process. The process X is said to be adapted to the filtration \left(\mathcal_i\right)_ if the random variable X_i: \Omega \to S is a (\mathcal_i, \Sigma)-measurable function for each i \in I. Examples Consider a stochastic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Wiener Space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Motivation Let H be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form :\frac\int_H f(v) e^ Dv, where Z is supposed to be a normalization constant and where Dv is supposed to be the non-existent Lebesgue measure on H. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. Physical interpretation of divergence In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation. The integration by parts formula states: \begin \int_a^b u(x) v'(x) \, dx & = \Big (x) v(x)\Biga^b - \int_a^b u'(x) v(x) \, dx\\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end Or, letting u = u(x) and du = u'(x) \,dx while v = v(x) and dv = v'(x) \, dx, the formula can be written more compactly: \int u \, dv \ =\ uv - \int v \, du. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. More general formulations of integration by parts ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malliavin Derivative
In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense. Definition Let H be the Cameron–Martin space, and C_ denote classical Wiener space: :H := \ := \; :C_ := C_ (, T \mathbb^) := \; By the Sobolev embedding theorem, H \subset C_0. Let :i : H \to C_ denote the inclusion map. Suppose that F : C_ \to \mathbb is Fréchet differentiable. Then the Fréchet derivative is a map :\mathrm F : C_ \to \mathrm (C_; \mathbb); i.e., for paths \sigma \in C_, \mathrm F (\sigma)\; is an element of C_^, the dual space to C_\;. Denote by \mathrm_ F(\sigma)\; the continuous linear map H \to \mathbb defined by :\mathrm_ F (\sigma) := \mathrm F (\sigma) \circ i : H \to \mathbb, sometimes known as the ''H''-derivative. Now define \nabla_ F : C_ \to H to be the adjoint of \mathrm_ F\; in the sense that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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