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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, 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] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predictable Process
In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. Mathematical definition Discrete-time process Given a filtered probability space (\Omega,\mathcal,(\mathcal_n)_,\mathbb), then a stochastic process (X_n)_ is ''predictable'' if X_ is measurable with respect to the σ-algebra \mathcal_n for each ''n''. Continuous-time process Given a filtered probability space (\Omega,\mathcal,(\mathcal_t)_,\mathbb), then a continuous-time stochastic process (X_t)_ is ''predictable'' if X, considered as a mapping from \Omega \times \mathbb_ , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes. This σ-algebra is also called the predictable σ-algebra. Examples * Every determini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyoshi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates. The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giulia Di Nunno
Giulia Di Nunno (born 1973) is an Italian mathematician specializing in stochastic analysis and financial mathematics who works as a professor of mathematics at the University of Oslo, with an adjunct appointment at the Norwegian School of Economics. As well as for her research, Di Nunno is known for promoting mathematics in Africa. Education and career Di Nunno earned a degree in mathematics from the University of Milan in 1998, including research on stochastic functions with Yurii Rozanov. She moved to the University of Pavia for doctoral studies, continuing with Rozanov as an informal mentor but under the official supervision of Eugenio Regazzini. She completed her Ph.D. in 2003; her dissertation was ''On stochastic differentiation with applications to minimal variance hedging''. She joined the University of Oslo in 2003, and added her affiliation with the Norwegian School of Economics in 2009. Activism Di Nunno is the chair of the European Mathematical Society's Committee fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernt Øksendal
Bernt Karsten Øksendal (born 10 April 1945 in Fredrikstad) is a Norwegian mathematician. He completed his undergraduate studies at the University of Oslo, working under Otte Hustad. He obtained his PhD from University of California, Los Angeles in 1971; his thesis was titled ''Peak Sets and Interpolation Sets for Some Algebras of Analytic Functions'' and was supervised by Theodore Gamelin. In 1991, he was appointed as a professor at the University of Oslo. In 1992, he was appointed as an adjunct professor at the Norwegian School of Economics and Business Administration, Bergen, Norway. His main field of interest is stochastic analysis, including stochastic control, optimal stopping, stochastic ordinary and partial differential equations and applications, particularly to physics, biology and finance. For his contributions to these fields, he was awarded the Nansen Prize in 1996. He has been a member of the Norwegian Academy of Science and Letters since 1996. He was elected as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Control
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. The context may be either discrete time or continuous time. Certainty equivalence An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that underlies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale Representation Theorem
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 with respect to \mathcal_t, such that :X = E(X) + \int_0^\infty C_s\,dB_s. Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
