View source for Stochastic calculus

{{Short description|Calculus on stochastic processes}}
{{No footnotes|date=August 2011}}
{{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. This field was created and started by the [[Japanese people|Japanese]] mathematician [[Kiyosi Itô]] during [[World War II]].

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 [[Itô calculus]] and its variational relative the [[Malliavin calculus]].  For technical reasons the Itô 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 Itô integral, and vice versa.  The main benefit of the Stratonovich integral is that it obeys the usual [[chain rule]] and therefore does not require [[Itô'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 Itô form.

== Itô integral ==
{{main|Itô calculus}}

The [[Itô 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 or Fisk–Stratonovich integral of a [[semimartingale]] <math>X</math> against another [[semimartingale]] ''Y'' can be defined in terms of the Itô 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 optional [[Quadratic variation|quadratic covariation]] of the continuous parts of ''X''
and&nbsp;''Y'', which is the optional quadratic covariation minus the jumps of the processes <math>X</math> and <math>Y</math>, i.e.
:<math>\left [ X, Y\right]_t^c:=
[X,Y]_t - \sum\limits_{s\leq t}\Delta X_s\Delta Y_s</math>.
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 [[mathematical finance]], in which asset prices are often assumed to follow [[stochastic differential equation]]s. For example, the [[Black–Scholes model]] prices options as if they follow a [[geometric Brownian motion]], illustrating the opportunities and risks from applying stochastic calculus.

== Stochastic integrals ==
Besides the classical Itô and Fisk–Stratonovich integrals, many other notions of stochastic integrals exist, such as the [[Skorokhod integral|Hitsuda–Skorokhod integral]], the [[Marcus integral]], and the [[Ogawa integral]].

==See also==
{{Portal|Mathematics}}
<!-- Please keep entries in alphabetical order & add a short description [[WP:SEEALSO]] -->
{{div col|colwidth=20em|small=yes}}
*[[Itô calculus]]
*[[Itô's lemma]]
*[[Stratonovich integral]]
*[[Semimartingale]]
*[[Wiener process]]
{{div col end}}
<!-- please keep entries in alphabetical order -->

== References ==
* Thomas Mikosch, 1998, Elementary Stochastic Calculus, World Scientific, {{ISBN|981-023543-7}}
* Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, {{isbn|9781848168312}}
* {{Cite journal|last1=Szabados|first1=T.S.|last2=Székely|first2=B.Z.|doi= 10.1007/s10959-007-0140-8|title=Stochastic Integration Based on Simple, Symmetric Random Walks|journal=Journal of Theoretical Probability|volume=22|pages=203–219|year = 2008|arxiv=0712.3908}} [https://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3908v2.pdf Preprint]

{{Industrial and applied mathematics}}
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[[Category:Stochastic calculus| ]]
[[Category:Mathematical finance]]
[[Category:Integral calculus]]
[[Category:Definitions of mathematical integration]]