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'''Stochastic calculus''' is a branch of [[mathematics]] that operates on [[stochastic process]]es. It allows a consistent theory of integration to be defined for [[integrals]] of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
In [[mathematics]], a '''delta operator''' is a shift-equivariant [[linear transformation|linear]] operator ''<math>\scriptstyle{ Q:\mathbb K[x] \longrightarrow \mathbb K[x] }</math>'' on the [[vector space]] of [[polynomial]]s in a variable <math> \scriptstyle x </math> over a [[field (mathematics)|field]] <math>\scriptstyle{ \mathbb K}</math> that reduces degrees by one.


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.
To say that <math>\scriptstyle Q</math> is '''shift-equivariant''' means that if <math>\scriptstyle{ g(x) = f(x + a)}</math>, then


The main flavours of stochastic calculus are the [[Itō calculus]] and its variational relative the [[Malliavin calculus]].  For technical reasons the Itō integral is the most useful for general classes of processes but the related [[Stratonovich integral]] is frequently useful in problem formulation (particularly in engineering disciplines.) The Stratonovich integral can readily be expressed in terms of the Itō integral. The main benefit of the Stratonovich integral is that it obeys the usual [[chain rule]] and does therefore not require [[Itō's lemma]]. This enables problems to be expressed in a co-ordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than '''R'''<sup>''n''</sup>.
:<math>{ (Qg)(x) = (Qf)(x + a)}.\,</math>
The [[dominated convergence theorem]] does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itō form.


==Itō integral==
In other words, if ''<math>f</math>'' is a "'''shift'''" of ''<math>g</math>'', then ''<math>Qf</math>'' is also a shift of ''<math>Qg</math>'', and has the same "'''shifting vector'''" ''<math>a</math>''.
{{main|Itō calculus}}


The [[Itō integral]] is central to the study of stochastic calculus. The integral <math>\int H\,dX</math> is defined for a [[semimartingale]] ''X'' and locally bounded '''predictable''' process ''H''. {{Citation needed|date=August 2011}}
To say that ''an operator reduces degree by one'' means that if ''<math>f</math>'' is a polynomial of degree ''<math>n</math>'', then ''<math>Qf</math>'' is either a polynomial of degree <math>n-1</math>, or, in case <math>n = 0</math>, ''<math>Qf</math>'' is 0.


==Stratonovich integral==
Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in ''<math>x</math>'' that maps ''<math>x</math>'' to a nonzero constant.  Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.
{{main|Stratonovich integral}}


The Stratonovich integral of a [[semimartingale]] <math>X</math> against another [[semimartingale]] ''Y'' can be defined in terms of the Itō integral as
==Examples==


:<math> \int_0^t X_{s-} \circ d Y_s : = \int_0^t X_{s-} d Y_s + \frac{1}{2} \left [ X, Y\right]_t^c,</math>
* The forward [[difference operator]]  


where [''X'',&nbsp;''Y'']<sub>''t''</sub><sup>''c''</sup> denotes the [[Quadratic variation|quadratic covariation]] of the continuous parts of ''X''
:: <math> (\Delta f)(x) = f(x + 1) - f(x)\, </math>
and&nbsp;''Y''. The alternative notation


:<math> \int_0^t X_s \, \partial Y_s </math>
:is a delta operator.


is also used to denote the Stratonovich integral.
* [[Derivative|Differentiation]] with respect to ''x'', written as ''D'', is also a delta operator.


==Applications==
* Any operator of the form
::<math>\sum_{k=1}^\infty c_k D^k</math>
: (where ''D''<sup>''n''</sup>(&fnof;) = &fnof;<sup>(''n'')</sup> is the ''n''<sup>th</sup> derivative) with <math>c_1\neq0</math> is a delta operator.  It can be shown that all delta operators can be written in this form.  For example, the difference operator given above can be expanded as
::<math>\Delta=e^D-1=\sum_{k=1}^\infty \frac{D^k}{k!}.</math>


A very important application of stochastic calculus is in [[quantitative finance]], in which asset prices are often assumed to follow [[stochastic differential equations]].  In the [[Black-Scholes model]], prices are assumed to follow the [[geometric Brownian motion]].
* The generalized derivative of [[time scale calculus]] which unifies the forward difference operator with the derivative of standard [[calculus]] is a delta operator.


{{No footnotes|date=August 2011}}
* In [[computer science]] and [[cybernetics]], the term "discrete-time delta operator" (&delta;) is generally taken to mean a difference operator


==References==
:: <math>{(\delta f)(x) = {{ f(x+\Delta t) - f(x) }  \over {\Delta t} }}, </math>


* Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, ISBN:9781848168312
: the [[Euler approximation]] of the usual derivative with a discrete sample time <math>\Delta t</math>. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.


*{{cite doi|10.1007/s10959-007-0140-8}} [http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3908v2.pdf Preprint]
==Basic polynomials==


[[Category:Stochastic calculus|*]]
Every delta operator ''<math>Q</math>'' has a unique sequence of "basic polynomials", a [[polynomial sequence]] defined by three conditions:
[[Category:Mathematical finance]]
[[Category:Integral calculus]]


[[ar:حساب التفاضل والتكامل العشوائيّ]]
* <math>\scriptstyle p_0(x)=1 ;</math>
[[de:Stochastische Integration]]
* <math>\scriptstyle p_{n}(0)=0;</math>
[[fr:Calcul stochastique]]
* <math>\scriptstyle (Qp_n)(x)=np_{n-1}(x), \; \forall n \in \mathbb N.</math>
[[gl:Cálculo estocástico]]
 
[[pt:Cálculo estocástico]]
Such a sequence of basic polynomials is always of [[binomial type]], and it can be shown that no other sequences of binomial type exist.  If the first two conditions above are dropped, then the third condition says this polynomial sequence is a [[Sheffer sequence]] -- a more general concept.
[[ru:Стохастический интеграл]]
 
[[uk:Теорія випадкових процесів]]
== See also ==
[[zh:随机分析]]
 
* [[Pincherle derivative]]
* [[Shift operator]]
* [[Umbral calculus]]
 
== References ==
* {{Citation | last1=Nikol'Skii | first1=Nikolai Kapitonovich | title=Treatise on the shift operator: spectral function theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-15021-5 | year=1986}}
 
 
[[Category:Linear algebra]]
[[Category:Polynomials]]
[[Category:Finite differences]]
 
[[pl:Operator delta]]

Revision as of 21:56, 10 August 2014

In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one.

To say that is shift-equivariant means that if , then

In other words, if is a "shift" of , then is also a shift of , and has the same "shifting vector" .

To say that an operator reduces degree by one means that if is a polynomial of degree , then is either a polynomial of degree , or, in case , is 0.

Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in that maps to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.

Examples

is a delta operator.
  • Any operator of the form
(where Dn(ƒ) = ƒ(n) is the nth derivative) with is a delta operator. It can be shown that all delta operators can be written in this form. For example, the difference operator given above can be expanded as
  • The generalized derivative of time scale calculus which unifies the forward difference operator with the derivative of standard calculus is a delta operator.
the Euler approximation of the usual derivative with a discrete sample time . The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.

Basic polynomials

Every delta operator has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:

Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a Sheffer sequence -- a more general concept.

See also

References

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pl:Operator delta