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Itō Calculus

Itō calculus

Itō calculus, named after Kiyoshi Itō, treats mathematical operations on stochastic processes. Its most important concept is the Itô stochastic integral. Before starting, it is important to note that:
- Capitalized letters such as X denote random variables.
- Capitalized letters with a subscript t such as B_t denote a stochastic process which is a set of random variables indexed by t.
- A small letter d to the left of a random process e.g. dB_t means an infinitesimal change in the random process which is a random variable. The stochastic integral of a process X_t with respect to a process B_t is denoted by :

\int_^ X_t\, dB_t

and is defined as the limit in probability of corresponding sums of the form :

\sum X_ (B_ - B_).

A crucial fact about this integral is Itō's lemma. Both summation and multiplication of random variables are defined in probability theory. The summation involves a convolution of the probability density function (pdf) and multiplication is repeated summation. Category:Stochastic processes Category:Equations

Kiyoshi Itō

Kiyoshi Itō (Japanese: 伊藤清, born September 7, 1915) is a Japanese mathematician, was born in Hokusei-cho, Mie Prefecture Japan. His name is often spelled Itô in the West, even though the correct Hepburn romanization is Itō. He introduced what is now called the Itō calculus, of which the basic concept is the Itō integral, and the most basic among important results is Itō's lemma.

External link

- Itō, Kiyoshi Itō, Kiyoshi Category:1915 births ja:伊藤清

Infinitesimal

In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. A number x ≠ 0 is an infinitesimal if every sum |x| + ... + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number. Infinitesimals, obviously, are not real numbers, so "operations" on them aren't familiar. The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence. See how Archimedes used infinitesimals. The Archimedean property is the property of an ordered algebraic structure of having no infinitesimals. When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go: ::To find the derivative f(x) of the function f(x) = x², let dx be an infinitesimal. Then, ::: ::since dx is infinitesimally small. This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The analyst: or a discourse addressed to an infidel mathematician. The fundamental problem is that dx is first treated as non-zero (because we divide by it), but later discarded as if it were zero. It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals. Thereafter, infinitesimals are numbers which represent limits or relationships between limits, and using them as symbols continues to be convenient for simplifying notation and calculation. Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of f(x) = x² can be justified with a minor modification: we have to talk about the standard part of the difference quotient, and the standard part of x + dx is x. Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., NOT (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above.
See also

- Hyperreal number
- Infinitesimal calculus
- Surreal number Category:Calculus ja:無限小

Itō's lemma

In mathematics, Itō's lemma is a theorem of stochastic calculus that shows that second order differential terms of Wiener processes become deterministic under stochastic integration. It is somewhat analogous in stochastic calculus to the chain rule in ordinary calculus. The lemma is widely employed in mathematical finance.

Statement of the lemma

Let x(t) be an Itō (or generalized Wiener) process. That is let :

dx(t) = a(x,t)\,dt + b(x,t)\,dW_t

where W_t is a Wiener process, and let f(x, t) be a function with continuous second derivatives. Then

f(x(t),t)

is also an Itō process, and :

df(x(t),t) = \left(a(x,t)\frac + \frac + \fracb(x,t)^2\frac\right)dt + b(x,t)\frac\,dW_t

Informal proof

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not handled carefully here. Expanding f(x, t) from above in a Taylor series in x and t we have :

df = \frac\,dx + \frac\,dt + \frac\frac\,dx^2 + \cdots

and substituting (adt + bdW) for dx gives :

df = \frac(a\,dt + b\,dW) + \frac\,dt + \frac\frac(a^2\,dt^2 + 2ab\,dt\,dW + b^2\,dW^2) + \cdots.

Reordering this to combine like differential terms, we have :

df = \left( a \frac + \frac\right)\,dt + \fracb^2\frac\,dW^2 + b\frac\,dW + e,

where e (an error term) is :

ab\frac\,dt\,dW + \fraca^2\frac\,dt^2 + \cdots.

In the limit as dt tends to 0, the dt² and dt dW terms disappear but the dW² term tends to dt. The latter can be shown if we prove that :

dW^2 \rightarrow E(dW^2),

since

E(dW^2) = dt. \,

The proof of this statistical property is however beyond the scope of this article. Substituting this dt in and reordering the terms so that the dt and dW terms are collected, we obtain :

df = \left(a\frac + \frac + \fracb^2\frac\right)dt + b\frac\,dW

as required. The formal proof, which is not included in this article, requires defining the stochastic integral, which is an advanced concept in between functional analysis and probability theory.

External links

- [http://www2.sjsu.edu/faculty/watkins/ito.htm Derivation], Prof. Thayer Watkins
- [http://www.quantnotes.com/fundamentals/backgroundmaths/ito.htm Discussion], quantnotes.com
- [http://www.ftsmodules.com/public/texts/optiontutor/chap6.8.htm Informal proof], optiontutor Category:Stochastic processes Category:Lemmas ja:伊藤の補題

Convolution

For the computer science usage see convolution (computer science) . ---- In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval.

Uses

Convolution and related operations are found in many applications of engineering and mathematics.
- In statistics, as noted above, a weighted moving average is a convolution.
- In statistics, the probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
- In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the blur circle formed by the iris diaphragm.
- In acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
- In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information).
- In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac delta function. See LTI system theory and digital signal processing.
- In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is sum of exponential decays from each delta pulse.
- In physics, wherever there is a linear system with a "superposition" principle, a convolution operation makes an appearance.

Definition

The convolution of f and g is written

f - g

. It is defined as the integral of the product of the two functions after one is reversed and shifted. :

(f   -  g )(t) = \int f(\tau) g(t - \tau)\, d\tau

The integration range depends on the domain on which the functions are defined. While the symbol

t

is used above, it need not represent the time domain. In the case of a finite integration range, f and g are often considered to extend periodically in both directions, so that the term g(t − τ) does not imply a range violation. This use of periodic domains is sometimes called a cyclic, circular or periodic convolution. Of course, extension with zeros is also possible. Using zero-extended or infinite domains is sometimes called a linear convolution, especially in the discrete case below. If

X

and

Y

are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the sum

X + Y

is given by the convolution f

-

g. For discrete functions, one can use a discrete version of the convolution. It is then given by :

(f   -  g)(m) = \sum_n  \,

When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, in this sense (using extension with zeros as mentioned above). Generalizing the above cases, the convolution can be defined for any two integrable functions defined on a locally compact topological group. A different generalization is the convolution of distributions.

Properties

The various convolution operators all satisfy the following properties:

Commutativity

f  -  g = g  -  f \,

Associativity

f   -  (g   -  h) = (f   -  g)   -  h \,

Distributivity

f   -  (g + h) = (f   -  g) + (f   -  h) \,

Associativity with scalar multiplication

a (f   -  g) = (a f)   -  g = f   -  (a g) \,

for any real (or complex) number

a

. Differentiation rule: :

\mathcal(f   -  g) = \mathcalf   -  g = f   -  \mathcalg \,

where Df denotes the derivative of f or, in the discrete case, the difference operator
Df(n) = f(n+1) - f(n).

Convolution theorem

The convolution theorem states that :

\mathcal(f   -  g) =  \mathcal (f) \cdot \mathcal (g)

where F(f) denotes the Fourier transform of f. Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform and Mellin transform.

Convolutions on groups

If G is a suitable group endowed with a measure m (for instance, a locally compact Hausdorff topological group with the Haar measure) and if f and g are real or complex valued m-integrable functions of G, then we can define their convolution by :

(f   -  g)(x) = \int_G f(y)g(xy^)\,dm(y) \,

In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the Peter-Weyl theorem of Harmonic analysis. It is very difficult to do these calculations without more structure, and Lie groups turn out to be the setting in which these things are done.

External links

-
- http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet. category:functional analysis Category:Image processing ja:畳み込み

Category:Stochastic processes

Category:Probability theory

Category:Equations

Category:Mathematics ja:Category:方程式 ko:분류:방정식

Raspai

Raspai és una entitat de població pertanyent al municipi de Iecla, en la Comunitat Murciana, i fronterera amb el País Valencià en la comarca del Vinalopó Mitjà. Té una població de 134 habitants, i el predomini lingüístic és del català. En l'actualitat s'estan impulsant, per part de l'Ajuntament de Iecla, cursos per a millorar la situació de la llengua catalana a Raspai i en entitats de població adjacents del Carxe. El prefix telefònic de la població és el d'Alacant, per proximitat. Categoria:Països Catalans Categoria:Regió de Múrcia

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