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Stochastic calculus is a branch of mathematics that operates on stochastic processes. 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.

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.

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 Rⁿ. 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

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The Itō integral is central to the study of stochastic calculus. The integral $\int H\,dX$ is defined for a semimartingale X and locally bounded predictable process H. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Stratonovich integral

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The Stratonovich integral of a semimartingale $X$ against another semimartingale Y can be defined in terms of the Itō integral as

\int _{0}^{t}X_{s-}\circ dY_{s}:=\int _{0}^{t}X_{s-}dY_{s}+{\frac {1}{2}}\left[X,Y\right]_{t}^{c},

where [X, Y]_t^c denotes the quadratic covariation of the continuous parts of X and Y. The alternative notation

\int _{0}^{t}X_{s}\,\partial Y_{s}

is also used to denote the Stratonovich integral.

Applications

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.

Template:No footnotes

References

Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, ISBN:9781848168312

Template:Cite doi Preprint

ar:حساب التفاضل والتكامل العشوائيّ de:Stochastische Integration fr:Calcul stochastique gl:Cálculo estocástico pt:Cálculo estocástico ru:Стохастический интеграл uk:Теорія випадкових процесів zh:随机分析

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-In [[arithmetic]], an [[Odd number|odd]] [[composite number|composite]] [[integer]] ''n'' is called an '''Euler pseudoprime''' to base ''a'', if ''a'' and ''n'' are [[coprime]], and
+'''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.
-: <math>a^{(n-1)/2} \equiv \pm 1\pmod{n}</math>
+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.
-(where ''mod'' refers to the [[modular arithmetic|modulo]] operation).
+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>.
+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.
-The motivation for this definition is the fact that all [[prime number]]s ''p'' satisfy the above equation which can be deduced from [[Fermat's little theorem]]. Fermat's theorem  asserts that if ''p'' is prime, and coprime to ''a'', then ''a''<sup>''p''&minus;1</sup> =  1 (mod ''p''). Suppose that ''p''>2 is prime, then ''p'' can be expressed as 2''q''&nbsp;+&nbsp;1 where ''q'' is an integer. Thus; ''a''<sup>(2''q''+1)&nbsp;&minus;&nbsp;1</sup> = 1 (mod&nbsp;''p'') which means that ''a''<sup>2''q''</sup>&nbsp;&minus;&nbsp;1 = 0 (mod ''p''). This can be factored as (''a''<sup>''q''</sup>&nbsp;&minus;&nbsp;1)(''a''<sup>''q''</sup> + 1) = 0 (mod ''p'') which is equivalent to ''a''<sup>(''p''&minus;1)/2</sup> = ±1 (mod&nbsp;''p'').
+==Itō integral==
+{{main|Itō calculus}}
-The equation can be tested rather quickly, which can be used for probabilistic [[prime testing|primality testing]]. These tests are twice as strong as tests based on Fermat's little theorem.
+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}}
-Every Euler pseudoprime is also a Fermat [[pseudoprime]]. It is not possible to produce a definite test of primality based on whether a [[number]] is an Euler pseudoprime because there exist ''absolute Euler pseudoprimes'', numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a [[subset]] of the absolute Fermat pseudoprimes, or [[Carmichael number]]s, and the smallest absolute Euler pseudoprime is [[1729 (number)|1729]] = 7&times;13&times;19.
+==Stratonovich integral==
+{{main|Stratonovich integral}}
-The stronger condition that ''a''<sup>(''n''&minus;1)/2</sup> =  (''a''/''n'') (mod ''n''), where (''a'',''n'') = 1 and (''a''/''n'') is the [[Jacobi symbol]], is sometimes used for a definition of an Euler pseudoprime. A discussion of numbers of this form can be found at [[Euler&ndash;Jacobi pseudoprime]].
+The Stratonovich integral of a [[semimartingale]] <math>X</math> against another [[semimartingale]] ''Y'' can be defined in terms of the Itō integral as
-==See also==
+:<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>
-* [[Probable prime]]
+where [''X'',&nbsp;''Y'']<sub>''t''</sub><sup>''c''</sup> denotes the [[Quadratic variation|quadratic covariation]] of the continuous parts of ''X''
+and&nbsp;''Y''. The alternative notation
+:<math> \int_0^t X_s \, \partial Y_s </math>
+is also used to denote the Stratonovich integral.
+==Applications==
+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]].
+{{No footnotes|date=August 2011}}
 ==References==
-*N. Koblitz, "A Course in Number Theory and Cryptography", Springer-Verlag, 1987.
+* Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, ISBN:9781848168312
-*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985.
+*{{cite doi|10.1007/s10959-007-0140-8}} [http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3908v2.pdf Preprint]
-{{DEFAULTSORT:Euler Pseudoprime}}
+[[Category:Stochastic calculus|*]]
-[[Category:Pseudoprimes]]
+[[Category:Mathematical finance]]
+[[Category:Integral calculus]]
-[[de:Eulersche Pseudoprimzahl]]
+[[ar:حساب التفاضل والتكامل العشوائيّ]]
-[[fr:Nombre pseudopremier d'Euler]]
+[[de:Stochastische Integration]]
-[[it:Pseudoprimo di Eulero]]
+[[fr:Calcul stochastique]]
+[[gl:Cálculo estocástico]]
+[[pt:Cálculo estocástico]]
+[[ru:Стохастический интеграл]]
+[[uk:Теорія випадкових процесів]]
+[[zh:随机分析]]

Main Page: Difference between revisions

Revision as of 21:18, 10 August 2014

Contents

Itō integral

Stratonovich integral

Applications

References

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Main Page: Difference between revisions

Revision as of 21:18, 10 August 2014

Itō integral

Stratonovich integral

Applications

References

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