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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] |
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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Wiener Space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener. Definition Consider ''E'' ⊆ R''n'' and a metric space (''M'', ''d''). The classical Wiener space ''C''(''E''; ''M'') is the space of all continuous functions ''f'' : ''E'' → ''M''. I.e. for every fixed ''t'' in ''E'', :d(f(s), f(t)) \to 0 as , s - t , \to 0. In almost all applications, one takes ''E'' = , ''T'' or , +∞) and ''M'' = R''n'' for some ''n'' in N. For brevity, write ''C'' for ''C''([0, ''T'' R''n''); this is a vector space. Write ''C''0 for the linear subspace consisting only of those function (mathematics), functions that take the val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itō Integral
Itō may refer to: *Itō (surname), a Japanese surname *Itō, Shizuoka, Shizuoka Prefecture, Japan *Ito District, Wakayama Prefecture, Japan See also * Itô's lemma, used in stochastic calculus *Itoh–Tsujii inversion algorithm, in field theory *Itô calculus, an extension of calculus to stochastic processes, named after Kiyoshi Itô *Ito (other) *ITO (other) Ito may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan * Itō, Shizuok ..., for the three-letter acronym {{DEFAULTSORT:Ito es:Ito fr:Ito nl:Ito ja:いとう pt:Ito ru:Ито ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girsanov Theorem
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a measure that is absolutely continuous with respect to ''P'' then every ''P''-semimartingale is a ''Q''-semimartingale. Stat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : '' absolutely continuous'' ⊆ '' uniformly continuous'' = '' ... [...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 stochasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A function that is ''not'' bounded is said to be unbounded. If ''f'' is real-valued and ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be bounded (from) above by ''A''. If ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be bounded (from) below by ''B''. A real-valued function is bounded if and only if it is bounded from above and below. An important special case is a bounded sequence, where ''X'' is taken to be the set N of natural numbers. Thus a sequence ''f'' = (''a''0, ''a''1, ''a''2, ...) is bounded if there exists a real number ''M'' such that :, a_n, \le M for every natural number ''n''. The set of all bounded sequences forms the sequence space l^\infty. The definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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