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{{Calculus |Specialized}}


'''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.


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.
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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 coordinate 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.
 
== Itō integral ==
{{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}}
 
== Stratonovich integral ==
{{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
 
:<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>
 
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 ==
 
An 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 ==
* Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, ISBN 9781848168312
* {{cite doi|10.1007/s10959-007-0140-8}} [http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3908v2.pdf Preprint]
 
[[Category:Stochastic calculus| ]]
[[Category:Mathematical finance]]
[[Category:Integral calculus]]

Latest revision as of 14:57, 1 July 2014


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