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Quadratic Variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued stochastic process defined on a probability space (\Omega,\mathcal,\mathbb) and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as t, defined as : t=\lim_\sum_^n(X_-X_)^2 where P ranges over partitions of the interval ,t/math> and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian motion. More generally, the covariation ... [...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] |
Bounded Variation
In mathematical analysis, a function of bounded variation, also known as ' 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 that the distance along the direction of the -axis, neglecting the contribution of motion along -axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed -axis and to the -axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob–Meyer Decomposition Theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer. History In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 Paul-André Meyer proved such a theorem, which became known as the Doob-Meyer decomposition. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.Protter 2005 Class D supermartingales A càdlàg supermartingale Z is of Class D if Z_0=0 and the collection : \ is uniformly integrable.Protter (2005) The theorem Let Z be a cadlag supermartingale of class D. Then there exists a uni ... [...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] |
Itô Isometry
In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals. Let W : , T\times \Omega \to \mathbb denote the canonical real-valued Wiener process defined up to time T > 0, and let X : , T\times \Omega \to \mathbb be a stochastic process that is adapted to the natural filtration \mathcal_^ of the Wiener process. Then :\operatorname \left \left( \int_0^T X_t \, \mathrm W_t \right)^2 \right= \operatorname \left \int_0^T X_t^2 \, \mathrm t \right where \operatorname denotes expectation with respect to classical Wiener measure. In other words, the Itô integral, as a function from the space L^2_ ( ,T\times \Omega) of square-integrable adapted processes to the space L^2 (\Omega) of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the ''Lp'' space with p=2. Among the ''Lp'' spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale. Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem). Definition Let (\Omega,F,P) be a probability space; let F_*=\ be a filtration of F; let X: adapted stochastic process on the set S. Then X is called an F_*-local martingale if there exists a sequence of F_*-stopping rule">stopping times \tau_k : \Omega \to [0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochasti ... [...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] |
Itô's Lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. Motivation Suppose we are given the stochastic differential equation dX_t = \mu_t\ dt + \sigma_t\ dB_t, where is a Wiener process and the functions \mu_t, \sigma_t are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution X_t directly in terms of B_t. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semimartingale
In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales. Definition A real valued process ''X'' defined on the filtered probability space (Ω,''F'',(''F''''t'')''t'' ≥ 0,P) is called a semimartingale if it can be decomposed as :X_t = M_t + A_t where ''M'' is a local martingale and ''A'' is a càdlàg adapted process of locally bounded variation. An R''n''-valued process ''X'' = (''X''1,…,''X''''n'') is a semimartingale i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
