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Itô Calculus
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: :Y_t=\int_0^t H_s\,dX_s, where ''H'' is a locally square-integrable process adapted to the filtration generated by ''X'' , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular ''t'' is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards". Substitution for a single variable Introduction Before stating the result rigorously, consider a simple case using indefinite integrals. Compute \textstyle\int(2x^3+1)^7(x^2)\,dx. Set u=2x^3+1. This means \textstyle\frac=6x^2, or in differential form, du=6x^2\,dx. Now :\int(2x^3 +1)^7(x^2)\,dx = \frac\int\underbrace_\underbrace_=\frac\int u^\,du=\frac\left(\fracu^\right)+C=\frac(2x^3+1)^+C, where C is an arbitrary constant of integration. This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by differentiating and comparing to the original integrand. :\frac\left frac(2x^3+1)^+C\right\f ... [...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] |
Local Time (mathematics)
In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields. Formal definition For a continuous real-valued semimartingale (B_s)_, the local time of B at the point x is the stochastic process which is informally defined by :L^x(t) =\int_0^t \delta(x-B_s)\,d s, where \delta is the Dirac delta function and /math> is the quadratic variation. It is a notion invented by Paul Lévy. The basic idea is that L^x(t) is an (appropriately rescaled and time-parametrized) measure of how much time B_s has spent at x up to time t. More rigorously, it may be written as the almost sure limit : L^x(t) =\lim_ \frac \int_0 ... [...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] |
Partition Of An Interval
In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that :. In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of . Every interval of the form is referred to as a subinterval of the partition ''x''. Refinement of a partition Another partition of the given interval , bis defined as a refinement of the partition , if contains all the points of and possibly some other points as well; the partition is said to be “finer” than . Given two partitions, and , one can always form their common refinement, denoted , which consists of all the points of and , in increasing order. Norm of a partition The norm (or mesh) of the partition : is the length of the longest of these subintervals : . Applications Partitions are used in the theory of the Riemann integral, the Riemann–St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Càdlàg
In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "continue à gauche, limite à droite", the left-right reversal of càdlàg, and càllàl for "continue à l'un, limite à l’autre" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let be a metric space, and let . A function is called a càdlàg function if, for every , * the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right-continuous
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 the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Random Variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The pair (''X'', Σ) is called a measurable space. A σ-algebra is a type of set algebra. An algebra of sets needs only to be closed under the union or intersection of ''finitely'' many subsets, which is a weaker condition. The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtered Probability Space
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter medium are described as ''oversize'' and the fluid that passes through is called the ''filtrate''. Oversize particles may form a filter cake on top of the filter and may also block the filter lattice, preventing the fluid phase from crossing the filter, known as ''blinding''. The size of the largest particles that can successfully pass through a filter is called the effective ''pore size'' of that filter. The separation of solid and fluid is imperfect; solids will be contaminated with some fluid and filtrate will contain fine particles (depending on the pore size, filter thickness and biological activity). Filtration occurs both in nature and in engineered systems; there are biological, geological, and industrial forms. Filtration is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clairvoyance
Clairvoyance (; ) is the magical ability to gain information about an object, person, location, or physical event through extrasensory perception. Any person who is claimed to have such ability is said to be a clairvoyant () ("one who sees clearly"). Claims for the existence of paranormal and psychic abilities such as clairvoyance have not been supported by scientific evidence. Carroll, Robert Todd. (2003)"Clairvoyance" Retrieved 2014-04-30. Parapsychology explores this possibility, but the existence of the paranormal is not accepted by the scientific community. The scientific community widely considers parapsychology, including the study of clairvoyance, a pseudoscience. Usage Pertaining to the ability of clear-sightedness, clairvoyance refers to the paranormal ability to see persons and events that are distant in time or space. It can be divided into roughly three classes: precognition, the ability to perceive or predict future events, retrocognition, the ability to see pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
